Hello Eric, if your possible half 2nd cousin were to test, you’d be looking at shared DNA of around 120 centimorgans (cM for short).
This would be quite a high level of cM to share with someone and would certainly indicate that you’re on the right track in terms of the closeness of connection. As an example, I have ten matches on Ancestry who are above 100cM and all are third cousins or closer.
Even if your possible half 2nd cousin doesn’t test, and you don’t have any matches above 100cM, you can still piece together connections with people who share lower levels of DNA and show (for example) that you’re directly related to the ancestors of X and Y’s father.
If you do test, just a personal view, but I’d go with Ancestry, partly because their DNA database is the biggest, and also because their algorithm is one of the most robust (some might say too robust!). You can also upload Ancestry DNA tests to other websites such as MyHeritage and FTDNA - although you can’t upload other DNA tests to Ancestry. The upload to MyHeritage and FTDNA is free but you have to pay to unlock their analysis tools (which Ancestry doesn’t have at all, and so something of a downside with this site). Then there are totally free sites (like GEDmatch) that you also might want to try.
For what it’s worth, I’d do the test. I spent years not finding my grandparents using just paper methods but tracked them down within a few weeks after getting the DNA results. And RootsChatters will always be happy to help you, either on the forum or through private message if matters get sensitive.
Good luck!