Repose en Paix is the exact equivalent you’re looking for. It is used the same way as “Rest In Peace”.

You could use ci-gît, meaning here lies and the name of the person. This can be used on gravestones.
Edit : I may not have answered your question right, I didn’t know what bereavement meant. Hope it will be useful to someone else.

If your question is how to refer to someone who passed away in a text or speech (rather than on a headstone), then traditional expressions are:

paix à son âme

que Dieu ait son âme

feu (as adjective, nowadays only used in legal context)

In modern French:

qui vient de nous quitter

qui nous a quittés

because of the circumstances – and the number of people at the service – I’d like to get it right!

Seeing your comment, it appears that what you should ask for is not “what is the translation of RIP ?” but “What should I say in this situation ?”. Beside you can find the translation of RIP easily in a dictionary.

If you want to get it rigth, and like Gilles hinted in the comment, you should focus on “the acceptability of religious phrasings or of the lack of religious phrasings” in your service. You migth be in a super religious and catholic office for all we know, but it is equally likely that you’re surrounded by atheist or pastafarian.

IMHO, you’d be better of using a neutral phrasing like Mes sincères condoléances.