I should have mentioned that I have also shot pointed jacketed bullets up to 220 gr. I don't think there's enough velocity to get them to expand, so they are probably a waste of time. Besides, they didn't shoot as well as the 180 gr XTP or the Beartooth cast bullet.
I suppose there's a possibility the 180 gr Hornady single shot pistol bullet would expand, but I doubt it would do as good as the XTP HP. The XTP will clock at about 1670 fps with the .360 DW load I was using in the group posted above. That's 1081 ft lbs of energy, which is pretty respectable.
The .357 Mag Handi has a very long throat, as you can see by the OAL I was using with .360 DW brass.
I tried a lot of combinations trying to get it to shoot right and one of the biggest factors to influence accuracy for me was to eliminate the roll crimp. When I started using a taper crimp (Lee dies will do a modified taper crimp if you set them right, but I use an extra taper crimp die) things began to work right. There really isn't a need for a roll crimp in a single shot and the taper crimp will provide plenty of bullet pull for good ignition. The Lee Factory Crimp Die may be a good option, but I haven't actually tried it to say for sure.
The 3 things I have done to change a bullet sprayer to a pretty good rifle are; 1) float the barrel 2) get the locking shelf right and 3) use a taper crimp.