Edit: The next paragraph was my original reply. However immediately after saving, I think what really did the trick was checkmarking the Consume Cut Bodies box in the Save Bodies feature. Since I mainly wanted the assembly I didn't mind since I can just build the part again "relatively" easily. The main issue was that the part was having issues saving, so the Consume Cut Bodies just eliminates having to save the bodies in the part file.
After a lot of fiddling I somehow got the feature to work. I used the Checking Model Geometry page in help.solidworks.com. It highlights a feature to "increase the level of error checking" by turning on Verification on rebuild in Tools>Options>Performance. I had used the Check Entity tool before but it wasn't catching any errors, and honestly, turning on Verification on rebuild didn't highlight any errors or complications either. But after running it, when I tried to use the Save Bodies feature again it was able to get through it all (after updating several times of course). So I ran with it, but would still like some sort of explanation if it can be provided.