Hi,
Basically, what you need to do is to build a program which will manage residual budget for you on particular fund and post budgetary documents, taking unconsumed budget from, say, fund April towards fund May. As you understand, there is no standard tool for this handling of the budget, because actual carry-forward works on annual basis; hence, it's irrelevant in your case.
You can use BAPI_0050_CREATE to generate these budgetary documents, but the program which runs the BAPI will have to calculate residual budget by itself looking, as correctly mentioned by you, on FMAVCT table.
Bottom line: feasible, but not standard and requires mid-scale development.
Regards,
Eli