How to help warp stabilizer focusing on the right object?
I use long workaround in Premiere and it works for all sorts of stabilizers not only for warp (I use 3rd party). 

What I do is masking out unwanted object. But you cannot do that on same clip, since Premiere will serve naked frame to the stabilizer tool. So what I do is put some effect on clip that masks moving parts. It can be title, power window, crop, or draw a mask in newer Premieres. Whatever suits you and track with a shot. 

Then you make Nested sequence out of this clip and put stabilizer on that. It will be served with image that is masked out. Not track clip. When done, just go back to original clip inside Nested and tick masking effect off (disable png, or whatever). It will be clean image again and properly done. If You need to change something and re-process it, just enable the underlying effect again. 

It is dirty and fiddly but that is how I did it well before Warp was added to premiere with other tools. I would like that would be implemented in the tool itself. Free tools on internet all have masking, but premiums do not bother to serve us better.