How can I help my advisor not lose face when changing from a thesis to non-thesis program?
Why are you worried about helping save face for this person?  It sounds like they have been nothing but problematic.  

If the department is OK with you switching to a non-degree program and graduating, just do it and get out of there.  Based on what you have reported, there seems to be no benefit for you doing otherwise: you're just making a lot of excuses for what is very poor behavior. For example, it doesn't matter if supervising your thesis is "low priority" and your advisor is "very busy"---supervising your thesis is part of your advisor's job.

Given that you are have already engaged both the program director and the Ombuds office and gotten their encouragement to proceed on this path, just go ahead and finish your degree without the thesis.  The only disadvantage you are likely to have is that you're not going to be able to get a good recommendation letter from your former advisor---but then, would you have been able to anyway?