Driving some screws through the thing would probably hold it together..

Hot glue? :p make sure it gets as hot as possible, put it in place of broken clips and assemble everything quickly (keep it squeezed together by putting a lot of force for couple of minutes). I did something like this with a laptop and it worked. You can warm it up with a hairdryer if you want to disassemble. It's not a cleanest solution, but it doesn't require modifying the case.

I decided I will try another approach. I will use epoxy glue and try to put the broken parts together very precisely one by one. The drying time is about 24 hours, so have to use adhesive tape to keep the parts intact.