As someone who also works with designers, I understand at a deep level how you end up in this situation—but I also don't think it should be this way. Image compression should be primarily an engineering issue, and designers should be giving you lossless or effectively lossless[1] images to compress as you see fit.

The one thing I'd designers to be responsible for is image resolution, since this is intrinsically tied to website layout and affects white source photos can be used where.

[1] For example, a 100% quality jpeg is perfectly fine to work with.