I am trying to repair a leaking bathtub faucet, probably an older mueller. I have a handle puller and tried it without any luck and now the puller won't come out of the center screw hole. Maybe I should try an pull it straight out, I tried to unscrew it. Ideas?

most handles pull off a spline, and most valve stems unscrew "open for more water" direction after the cap is removed.

if these parts aren't moving, the doctor recommends you take two shots of penetration oil on them, with a light wrench whack, in 5-minute intervals, and check them in the morning. repeat as necessary.

I've never once seen any handle that unscrews, the only things that unscrew are the screw holding it on and the stem.
When I use a handle puller I loosen the screw a few turns, not take it all the way out and use the center of the phillips head screw to center the puller. All the puller has to do is break the knob free. Once it starts to move it will pop right off without the puller attached once the screw is completly removed.
If you push againt the stem it can mess up the threads and the shaft on the stem.