Stuff that “works” but I’m not happy with
There are some massively overpowered approaches that can do this (and so much more) that aren’t really the kinds of answers I was looking for.
Artificer cheese, Persist cheese
The artificer (Eberron Campaign Setting) is a problematic class, in E6 especially. The ability to craft items that require 2 caster levels more than the artificer actually has is always potent, but it’s ludicrous in E6 where it means you can trivially craft spell-completion/spell-trigger items of 4th-level spells, so long as you have caster level 5th or greater.
On top of that, the 3rd-level artificer infusion metamagic item (Eberron Campaign Setting)—available to any 5th-level artificer—just allows you to add any metamagic feat to any spell-trigger item, and cast the spell with that metamagic freely added to the spell for the next few rounds. When that metamagic feat is Persist Spell, you get incredible value out of this. And among non-artificers, Divine Metamagic, Metamagic Song paired with Talfirian Song, and so on, all the various ways to cheese Persist Spell onto things at low levels, still apply to allow any buffs we want to stay up all day, allowing us to craft. Metamagic item is by far the easiest of them, though.
So that means an artificer can trivially persist spells, and they can be up to 4th-level, without any real effort. Among 4th-level spells is consumptive field (Spell Compendium), suggested by fectin in a now-deleted answer, which is another broken option that works like death knell except you can build up lots of “deaths” to pump your CL. We only need 6, which is not difficult to achieve.
Even without consumptive field, though, there are a lot of 3rd-level and 4th-level spells that boost caster level a bit, and the artificer can easily make sure they last. So can a cleric, though they’ll be limited to the 3rd-level spells (or possibly 4th-level spells found on their domains).
This approach works, but isn’t what I was looking for.
Stuff that might work, and I might be OK with, but lots of questions
The following is an approach I don’t love, but would prefer over artificer and Persist cheese. More importantly, though, the actual rules here are very unclear, so it’s not certain that it actually works.
Master Spellthief cheese
The Master Spellthief feat (Complete Scoundrel) allows your arcane caster level in each arcane class you’re part of to have the caster level of the sum total of all arcane caster levels you have. Bonuses to caster level shouldn’t get “double counted”—it refers to levels—but noctumancer (Tome of Magic) and ultimate magus (Complete Mage) can advance two arcane spellcasting (mystery-using) classes at once. We would need early entry shenanigans to actually benefit from either of these, and even then Master Spellthief is a really questionable, unclear feat.
I may return to this to see how it works out, but for right now this is filed under “too many questions, too unclear.”
Stuff that I’m happy with, but doesn’t quite get to CL 12th
If we assume, for the sake of sanity, that artificers are prevented from creating items of 4th-level spells in E6, that metamagic item is banned or nerfed in some fashion to make it less blatantly broken, Persist Spell is... overlooked as an unpalatable option, and consumptive field is just banned as a bad idea, we’re left with a much more difficult time. As of yet, I still haven’t found a way to get there.
First, Practiced Spellcaster can be used to ensure we have caster level 6th in any caster class we have at least two (effective) levels in. It can’t get us beyond 6th, but it’s helpful for allowing us to, say, have multiple types of spellcasting capable of getting there. We can assume that no matter what, we’ll always have 6 caster levels we need to make up.
Various prestige classes have a “spell power” class feature, which gives +1 to caster levels. These are either high level or else restricted to a certain class of spells, but the ultimate magus’s version appears at 1st level (ECL 6th), and applies to all arcane spells (Complete Mage). One can qualify for ultimate magus as a 1st-level wizard or wu jen that takes Alacritous Cogitation (Complete Mage) and Precocious Apprentice (Complete Arcane). Ultimate magus also helpfully supplies a second level of spellcasting for that class, allowing you to take Practiced Spellcaster to get to caster level 6th. Unfortunately, only works for arcane caster levels.
Membership in a church that offers the Magic domain can, per Complete Champion, grant “Effective caster level for a single type of magic (arcane of a single school or divine, your choice) increases by 1” at 23 affiliation. Affiliation isn’t generally hard to get, though 23 is quite a lot of it. Still, not prohibitively difficult. If nothing else, going on a quest to “[destroy foes] single-handedly using only magic,” will get you there eventually.
Boosting your caster level for a single school of arcane magic probably doesn’t meet the requirements for Craft Staff, but boosting the caster level of all of your divine magic probably does. So this is only useful for Craft Staff if we’re a divine spellcaster, which frustratingly means it can’t combine with ultimate magus.
Draconic Power (Races of the Dragon) gives a +1 bonus to caster level—and it’s not restricted to arcane spells the way the saving throw DC bonus is. It requires Draconic Heritage (Races of the Dragon), which requires being a 1st-level sorcerer, but if desired we can avoid that with the Dragontouched feat (Dragon Magic). But since we’re already interested in ultimate magus, it’s worth noting that sorcerer can also save us a feat on Alacritous Cogitation. Three feats, or a level and two feats, is a ton, but it counts.
Inspire Spellpower (Races of Stone) can be taken by a bard with 8 ranks in Perform in order to give other spellcasters that can hear them a +1 bonus to caster level. Necessitates another ally being involved. Like all bardic music effects, lasts as long as you can keep playing, so a construct or undead bard is ideal.
There is a 2nd-level sor/wiz Conjuration (Creation) spell, create magic tattoo (Player’s Guide to Faerûn), that can grant +1 caster level for 24 hours, i.e. we don’t need Persist Spell. Unfortunately, we need caster level 13th to enable the +1 caster level option. However, since we can benefit from various buffs that apply only to Conjuration and/or Creation spells, or that last a limited time (note that create magic tattoo takes 10 minutes to cast), to get there, making it more approachable than actual crafting.
The Craft and Creation domains (Spell Compendium) each gives +1 to caster level with Conjuration (Creation) spells. We can easily dip cleric for those.
Reserves of Strength (Dragonlance Campaign Setting) can grant a spell a bonus to caster level of up to +3 so long as we’re OK with being stunned for an equal number of rounds after casting it. And we’re definitely OK with that. It requires Iron Will, but that can be gotten for 3,000 gp at Otyugh Hole (Complete Scoundrel) if desired. (Reserves of Strength is a contentious feat, but this usage doesn’t get into any of the problems it can have.)
Any two of Magic domain church affiliation, Draconic Power, Inspire Spellpower, and/or ultimate magus gets us the rest of the way to 13th.
So we can cast create magic tattoo with high enough caster level to enable the +1 bonus to caster level option. It does require a cleric/wizard/ultimate magus that also needs either a level of sorcerer, two more feats, or the assistance of a bard.
Artificer still plays a role, as does its +2 bonus to caster level for items’ prerequisites, but we’re avoiding 4th-level spells and Persist Spell. Instead, we have the fact that, at caster level 10th, an artificer can craft an orange ioun stone for +1 caster level. That gets us one of the caster levels we need, but only if we can get the artificer to caster level 10th—and ultimate magus won’t help (though Draconic Power, Inspire Spellpower, and create magic tattoo will).
So we can get an artificer to caster level 9th, and we can get an arcane spellcaster to caster level 10th. If we can figure out how to get the artificer to caster level 10th, we can also push the arcane spellcaster to caster level 11th, and if whatever we use to get the artificer to 10th works for the arcane spellcaster, we can also get the arcane spellcaster to 12th, and they will be able to take and use Craft Staff. But we still need one more caster level somehow, so this doesn’t work.
(Much thanks to my friends Eurus and Linae [@forrestfire here], who suggested many of these options. Also thanks to @fectin, who discovered the Magic domain church affiliation.)