This question is regarding whether temping is culture-specific term. Isn't this a single-word question belonging on English Language & Usage? I am quite unsure, since no-one mentioned this in a comment under the question. If it does belong here, is there a part of Writing I am unaware of?
On its own, it's a question for English.
But the longer question does connect to a fictional setting and it's reasonable to ask in a writing context. If someone new to SE were coming in with that question asking where to post it, I'd probably direct them to English. But given the story context, it's not unreasonable to have it on Writing.
I have a character working in a short-term temporary position in an organisation and refer to her doing the work using the usual UK word, 'temping'. Will this be understood by readers in the US and other English-speaking cultures? If not, is there an equivalent culture-neutral term or should I spell it out?
I don't think that the question ever belonged to ELL. In fact, it seems to me that the OP already knows the meaning and usage of the word.
The question is about whether the use of a certain word would be understood by the readers in the context of a written text, and, if not, how to rewrite it in a way that it would be understood. In this sense it is akin to the discussion about the Aluminium/Aluminum conundrum.