Two daft ideas from myself
if you pop your multibody part into an assembly, you can zero offset (copy) surfaces to a new part which you can then save out and insert into your assembly.
A dafter idea (that works, I tried) is to thicken your surfaces and then in the split part either copy surface/delete body or just delete faces.
If you can repair or recreate the surfaces back into viable solids, this will work.
I know this is a worthless suggestion when there are eight thousand import errors. But if there isn't too many problem parts, you can work with the surfaces to try to get them to knit back together. Or recreate the solid if it isn't that difficult.
You can manually save the assembly as different file names and delete everything in it except the entities that relate to the body you are trying to salvage. Tedious but sometimes effective.
Did you do import diagnostics and let the software fix what it can? Dangerous at times because sometimes it's fix is irrational. But often most of your errors will go away.
Can you post the original file?
What version are you on? 2017 is reported to have better import capabilities including native formats from competitor CAD software. Maybe worth a try if you're not on 17 to put it on a separate computer for importing. Then if it does work, you can export as a parasolid to open a clean file with your version.
Because of ZTG errors, some errors can not be fixed on failed surfaces without direct manipulation. There are third-party addins that are designed to expedite this manual work.
Here's one (no affiliation):
and there are others as well. Also, if you check with capvidia, if you don't import often, they can clean up a file for you on a consulting basis. Probably a lot cheaper than buying the software.