I think you want to remove the bounded condition. I don't see how the category of chain complexes with bounded cohomology could be a model category. It doesn't have all small colimits; just take longer and longer chain complexes with trivial differentials, and you get something with unbounded cohomology.

I think you don't want any bounded condition. I don't see how the category of chain complexes with bounded cohomology could be a model category. It doesn't have all small colimits; just take longer and longer chain complexes with trivial differentials, and you get something with unbounded cohomology.