What is the word for "to do something one considers to be beneath oneself"?

The verb deign means to do something you consider to be beneath your dignity. It doesn't exactly fit in your example sentence - you would use it like

She finally deigned to pick up the trash bag...

Google Books claims 2220 written instances of...

People who] would not stoop to that

stoop (definition 2) - lower one’s moral standards so far as to do something reprehensible

It's a figurative extension from the primary meaning bend one’s head or body forwards and downwards (i.e. - metaphorically drop to a "lower" level of moral standards than normal).

You can use the verb condescend to get the message across:

She finally condescended to pick up the trash bag and fling it into the waste bin.

The verb is often used to show disapproval, so it should fit your example. Definition:

to do something that one regards as below one's dignity

(Collins Dictionary)

If someone condescends to do something, they agree to do it, but in a way which shows that they think they are better than other people and should not have to do it.

(Collins Dictionary for Learners)

Consider,

She swallowed her pride as she finally picked up the trash bag...

swallow one's pride: fig. to forget one's pride and accept something humiliating. McGraw-Hill Dictionary of American Idioms and Phrasal Verbs

She finally lowered herself to picking up the trash bag...

lower oneself: to humble oneself; to do something one considers to be beneath one's dignity. Your Dictionary

She got off her high horse as she finally picked up the trash bag...

get off one's high horse and get (down) off one's high horse

: to become humble; to be less haughty. It's about time that you got down off your high horse. Would you get off your high horse and talk to me? McGraw-Hill Dictionary of American Idioms and Phrasal Verbs

If you go from a slightly different angle (and don't mind a tinge of archaism), you could use abase.

TheFreeDictionary gives its definition as follows: