Bonus Feats are usually a class feature
So to get some, you need to take classes that have them.
Major Exception: Humans, and some human subraces, get a free Bonus Feat. This is why Human is usually the best choice for pretty much everything.
The only other exceptions I’m aware of off the top of my head are the Elder Evil’s rules for worshiping an elder evil, which gets you some bonus feats, and then spells that temporarily mimic feats, including some like mirror move and heroics that allow you a lot of choice in the feats you get.
There’s also the Dark Chaos Feat Shuffle (DCFS), which involves casting embrace the dark chaos (switch any feat you have for any Abyssal heritor feat) and then casting shun the dark chaos (switch any Abyssal heritor feat you have for any other feat), both from Fiendish Codex I: Hordes of the Abyss, but this doesn’t get you extra feats, just lets you trade useless feats for useful ones. For example, elves can technically trade away their Martial Weapon Proficiency feats. It’s also usually regarded as inappropriate for most tables.
Since Elder Evil, Fiendish Codex I, and Spell Compendium are not on your list, you’re not going to be able to use any of these anyway.
Useful class for you: Master Specialist
Sadly, there is a perfect class for you, but it’s in Complete Mage. The Master Specialist can be entered at level 4 by a specialist wizard who has the Spell Focus feat for their school, and gives Skill Focus (Spellcraft) as a bonus feat at 2nd level. So you could be Cleric 1/Wizard 3/Master Specialist 2, and have most of the feats you need for Archmage. That’s also 6 levels, so it’ll cost you a level of Archmage at the end, but that’s pretty minor.
Many 3.5 games, as a houserule, as well as Pathfinder, as an official rule, give out feats every odd level, instead of every third level. Could ask your DM to consider it.
Unearthed Arcana, which is available free as part of the SRD, also has a number of options that are useful to you.
The Martial Wizard variant gets a Fighter Bonus Feat instead of Scribe Scroll. You may be able to pick up one of the feats you need from that list, perhaps.
Further, you might ask your DM to consider the Flaws variant, which allow you to get up to two extra feats by taking some drawbacks. These drawbacks don’t usually really deserve a feat, so in a lot of ways it is always a good trade to take flaws for feats, which is a problem with the system. On the other hand, many feel that 3.5 gives out too few feats as it is, especially at low levels, which the Flaw system does fix.