The first thing I'd try is to just close SolidWorks and open it again and see if it is back to normal.
Done that. I actually restarted my computer.
There is a possibility that some pop up window is there which might be out of your screen or hiding behind. And until you click on that, SW seems freezed. Try inserting a new part inside the assembly using Insert > Component > New Part and make sure you save it internally in assembly. If this seems OK then try inserting already available part and see what happens.
If you are seeing any mate errors, a good tool to help resolve them is "Tools>MateXpert" - but I don'tr think you're having that issue today...
I would leave your offending component in place (so that you are locked) and suppress mates one by one - starting from the bottom of your tree to see which mate is caiusing your trouble. It sounds to me that something is over defined in a way that SWX is not complaining. It may be that the assembly is behaving properly - it is just that the combination of mates is locking you up.
If you are able to isolate which mate is causing a lock up, then you need to scrutinize all of the constraints you've made and decide how to get arounsd the conflict.
I could be wrong and it could be a file corruption - did you try creating a brand new part that is "similar" to the last one and mat it in the same way to see if it is repeatable?
I re-read your OP and your last paragraph makes me think that there is something going on with the overall mate scheme. Is it a crazy suggestion to just delete ALL your mates and reconstruct the whole assy - or at least the major components and leave the hardware alone...
I am intrigued and would love to poke around.. Surely it is too large to pack-n-go?
Can you ZIP the assy and all the parts and post here so we can see what's going on?