Glad you found a free state that accepts refugees!
My preference for the Speer 135 grain +P load over my former favorite (and still #2 choice) FBI load (158 grain LSWCHP +P) for .38 Special snubs comes from the following, not necessarily in order of priority.
1. Excellent track record with NYPD and other large depts.
2. Superior accuracy in my experience to LHP +P .38 out of snubs in particular.
3. Palpably lighter recoil.
4. As you've noted, in very light revolvers (such as S&W 342, one of the guns in my own "carry rotation" for backup, or the 340 PD), I've seen the lead bullet loads "prairie dog" forward sufficiently to protrude from the front of the cylinder and bind the gun. You saw it in your heavier 340 M&P. The longer cylinder plus the greater weight of the 340 M&P over the 340 PD has, in my experience, kept it from moving far enough forward to lock up THAT gun, but it's still unsettling. The much tighter crimp on the Gold Dot jacketed bullet seems to always keep this from happening, in my experience.
The milder .357 loads with jacketed bullets are very promising in the little guns, but still much snappier in terms of controllability than the 135 grain Speer Gold Dot .38 Special +P. For me, the latter is an ideal balance of downrange performance with in-hand controllability. Of course, YMMV.