Tips for packing electrical cables near Giant Quick-E bottom bracket
Cable ties/zip ties can add bulk.  You might be able to recover some volume by changing them out for thin tie wire, looped exactly the same way.

Second thought - there's a lot of wire in there.  Can you shove any of it back into the tubes from which it comes?

Lastly, there are some black cables on the lower-left of your image.  Looks like they might tuck into a recess in the underside of the housing?    Try to methodically back-fill starting at the least accessible wiring, and then work outward covering your first efforts.    

TBH it looks like a difficult task, and I don't envy you.  Work with lots of illumination, and perhaps a selection of push-tools and pliers.  Take your time, and see what you can achieve.

And if it is just not going together, shake it harshly while pressing the cover closed, exactly like the factory did!