"Knapsack problem" with repetition, "lesser or equal" constraint, and recording all valid combinations
The number of solutions can easily be huge, so just writing out the solutions will take a long time. But this is not the knapsack problem, which gives you a budget (knapsack capacity) and values of the items, and ask for maximal value (or if some value is atainable).
One way to generate them all is to use backtracking: List all solutions that don't include object 1, then all that do include object 1 once, twice, ... Recursively check for solutions without/with object 2, ... I don't think there can be any more efficient way, and it does prune off combinations that are doomed (can't be extended).
Look if you can narrow the search somehow. Say order the objects so that you get promising combinations early, and cut off the search when you get enough of those.