Bin-packing satisfiability rather than minimization
Having a fixed number of bins makes little difference. If you use sorted-first-fit for example, the only difference is that the algorithm can fail. On the other hand, if you use a better algorithm for bin-packing (one that considers all bins before it is proven that they are all needed), you would find a trivial lower bound l for the number of bins, try to fit them all, and if it fails increase l by 1 and start all over again. 

The different sizes make the problem likely harder to solve. For example, if you only wanted to decide which bin contains the largest item, that would in normal bin packing be no problem: Just put it into the first bin; it doesn't matter which one. With different bin sizes, pick all the bins that are big enough, remove duplicates with the same size, and try putting it into each of these bins. So you have more possible combinations. (If you had 100 bins of size 10 and 100 bins of size 5, then items of size 6 would be no problem, but a largest item of size 5 or smaller would be put into one of two bins). 

I say "likely harder" because in the extreme you might have items that fit into one bin only. 

Once you try to find an optimal solution using brute force, any additional restrictions make it easier because they remove possibilities from the search tree.