Can Use Magic Device be used to bypass the spell on spell list portion for recharging staves?
Yes
to your specific question: You can in fact bypass the requirement to have one of the spells on your class' spell list with Use Magic Device, as you have quoted the relevant section yourself already.
Does this extend to ignoring the spell-list requirement for recharging a staff (as long as the character in question has the
required spell slot), or does that not count as "using" it in this
context?
This spell-list requirement for recharging a staff most certainly counts as "using" the staff, because this is the literal heading of the entire section:
Staves
Using Staves: […] Each morning, when a spellcaster prepares spells or regains spell slots, he can also imbue one staff with a portion of
his power so long as one or more of the spells cast by the staff is on
his spell list and he is capable of casting at least one of the
spells. Imbuing a staff with this power restores one charge to the
staff, but the caster must forgo one prepared spell or spell slot of a
level equal to the highest-level spell cast by the staff. […]
But…
No
you cannot use Use Magic Device to recharge a staff unless you are capable of casting one of the staff's spells.
Use Magic Device does not allow you to bypass the requirement to be "capable of casting at least one of the spells." This statement of yours:
This presents to prerequisites to recharging a staff: A, one or more
of the spells in the staff are on their spell list and they can cast
it; and B, they have a prepared spell or spell slot of a level equal
to the highest-level spell cast by the staff.
has to be amended. The prerequisites to recharging a staff are as follows:
- A: "one or more of the spells in the staff are on their spell list"
- B: the caster is capable of casting at least one of the spells
- C: "they have a prepared spell or spell slot of a level equal to the highest-level spell cast by the staff"
A and B are not the same requirements, and Use Magic Device can only overcome A. This is clear from the rules texts themselves (RAW), but also from the way they are meant to work: Wands, staves, and other spell-trigger items do not have the requirement to be able to cast one of the spells inside them, so this part of Use Magic Device does not intend to overcome such a requirement (RAI).
However, if you are capable of casting one of the staff's spells, for example through a spell-like ability, and you have a spell-slot of the required level to boot, then you can indeed recharge the staff.
Requirements A and B also do not necessarily refer to the same spell, so having one spell to satisfy requirement A (for example through Use Magic Device) and another spell to satisfy requirement B is legitimate.
Spell-like abilities can be used to overcome the B requirement, because the staff specifically calls out particular spells explicitly [the spells the staff is capable of casting], as per this FAQ:
Spell-Like Abilities, Casting, and Prerequisites: Does a creature with a spell-like ability count as being able to cast that spell for
the purpose of prerequisites or requirements?
Only if the pre-requisite calls out the name of a spell explicitly. For instance, the Dimensional Agility feat (Ultimate Combat) has
"ability to use the abundant step class feature or cast dimension
door" as a prerequisite; a barghest has dimension door as a spell-like
ability, so the barghest meets the "able to cast dimension door
prerequisite for that feat. However, the barghest's dimension door
would not meet requirements such as "Ability to cast 4th level spells"
or "Ability to cast arcane spells".
