| | Re: reimburse/recompense/compensate
None of them, a) because they're all much too formal for a conversation with friends; b) because the prepositions are wrong; c) because of their meaning :
1) reimburse : I suspect this is what was intended, but it's not accurate. Reimburse means to pay someone back when they have already spent money on things they aren't responsible for - eg a company reimburses their employee's travelling expenses after business trips. This is not the case here, as the flatmate is responsible for paying for the A/C, at least some of it. Peter may be expected to contribute to the payment, but not to reimburse the whole sum. Notice the structure incidentally : to reimburse (somebody for) something.
2) recompense : pay money to someone because you have harmed them in some way . Eg John and Peter might tell their landlord : Don't worry - we'll recompense you for all the damage that was caused during the party. Again - the preposition is for
3) compensate : similar to recompense but with a wider use, often unconnected with money and just meaning "make up for in some way". Eg: She spent the rest of her life trying to compensate for not having been with her children when they were young / No amount of money can compensate for his death. And again - compensate for
So - none of the verbs fit in terms of meaning, they're much too formal for the context, and the prepositions are wrong. What Peter would actually say would be something like : I'm sorry that I can't contribute to /towards the cost of the a/c. or I'm sorry that I can't give you something towards the cost of the a/c. or I'm sorry that I can't share the cost of the a/c.
If he could afford it, he might say - eg You pay the bill when you're in town, and I'll reimburse you later but again it's too formal (better : You pay the bill... and I'll pay you back later) and suggests he's going to pay all the money, not just his share.