A word that refers to saying something in an attempt to anger or embarrass someone into doing something that they don’t want to do?
In addition to Marcellothearcane's excellent answer I suggest goad.
From Lexico:
Goad
VERB
- Provoke or annoy (someone) so as to stimulate some action or reaction.
‘He was trying to goad her into a fight.’
The word derives from a spiked stick used to push cattle along. This definitely provides that same subtext that you're seeking to add.
The word that springs to mind is provoke:
- stimulate or give rise to (a reaction or emotion, typically a strong or unwelcome one) in someone.
- deliberately make (someone) annoyed or angry.
Lexico
It is used to signify a successful provocation (when someone responds), and also the act of provoking, which may or may not yield results:
- ‘Sometimes applicants are deliberately provoked to see how they handle themselves.’
- ‘Rachel refused to be provoked’
Synonyms include incense, enrage, irritate, infuriate, exasperate, exacerbate, madden, pique, nettle.
Another option would be to 'needle' someone into doing something:
- incite to action by repeated gibes e.g. "needled the boy into a fight"
[https://www.merriam-webster.com/dictionary/needle]
I think "incite" can also work in your context -
Incite
: to encourage someone to do something violent, illegal, or unpleasant, especially by making them angry (or excited).
Example sentences -
The rock band's failure to show up incited a riot, as the crowd had waited for hours.
He incited the workforce to go on strike.
(Source)
How about bait?
- b. entice, lure