I commend you for doing a great job of explaining it. Very clear.
However, while I can picture what you are doing, there might be more clues if we can see the assembly/parts.
Can you share an assembly that is exhibiting this behavior (even if it is a sample file)? I am on SW 2017, so I wouldn't be able to open it, but someone here should be able to. In addition to this, can you share a screen shot or two.
There can be several reasons for the issue you're seeing, so as Dan has mentioned files or some screen shots would be help.
In the meanwhile can you try these two things:
1. Make a copy of the assembly and open/check that for issue.
2. Insert this assembly into a new assembly and check that for issue.
Also it might be related to SPR 1081551, so report to your VAR.
Here are some screen shots.
One detail I accidentally left out is that I can create the desired cut-away view by suppressing and unsuppressing the proper components and cuts in the right order, but as soon as I begin to render the vessel, it always reverts to the visual shown in Iso view - Cuts Unsuppressed.
Iso View - Cuts Unsuppressed.
Top View - Cuts Suppressed
Top View - Cuts Unsuppressed
Desired Cut-Away View
It looks like this cylinder has a series of little holes. How are these holes defined?
Honestly, what it looks like to me is that this is the dotted line that is made when you use the hole wizard to create a threaded hole. It looks like it may be being left behind.
Try turning off the viewing of top level or component level annotations and see if that helps:
The holes in the cylinder are simply a circular pattern of a curve driven pattern of an extruded cut. It needs to be there. The image you copied to your reply is the image I would like to render (but with the full vessel). The top image (Iso View - Cuts Unsuppressed) is the one that it keeps reverting to randomly before I can render it.