I think it is obvious that Julia's comments were not intended as a bribe. OTOH, it is clear to me that her intention is to give up her stipend. I would assume her purpose in such an action is to show that she's not doing it for the money and would prefer to see the money go back to the voters.
I do find it interesting that no where in the letter to Julia does is state that she ever linked giving this money away to voters (any voter) to a vote for her. I am having trouble figuring out how that is considered a bribe for someone's vote. Of course, even if Julia was trying (which she's not) to tie the $ to a vote for her, since we have strict voter privacy, Julia could never know for certain how any voter actually voted (which is a good thing).
I think I would want to know how they see this as a bribe when no talk of "vote for me and you might win" happened.
It may be possible to figure out a different way to give away this money, that won't cause one of your antagonists to run to mommy.
You could refuse to fill out the paperwork to get the money, thereby leaving the money in the general fund since it would be budgeted, but not spent on your stipend.
You could fill out the paperwork and donate your money to various local charities. If you want to prove that you are not keeping any of the money for yourself, you could always scan your "paychecks" and then the receipt you get from the various organizations you donate the $ to.
You could fill out the paperwork and spend the money on a project to improve the city of Keene.
Just a few ideas - I'm sure there are other, better ones, as well.