Thanks for the suggestion of running it from the Terminal. AVL Looms should just admit it's 'Windows only' instead of selling their software as a Mac-capable solution. If I was the one to use it I would probably figure it out, but I can't expect non-technical end users do deal with launching stuff from the Terminal and dealing with a very un-Mac-like UI. This time the errors were still present but Wine did run, and WeavePoint ran after a fashion. Then I quit POM and ran it one more time.
The second run it did a lot in the Terminal with errors about recognized compression schemes, unfound libraries, failed dll warnings, etc (I can supply the Terminal output if you really want to read all that).
So I quit POM and grabbed XQuartz 2.7.11 and installed it. The first issue was that it wanted XQuartz. The Terminal had a lot of activity, lots of errors. $ /Applications/PlayOnMac.app/Contents/MacOS/playonmac There are a few ways to launch from the Terminal.