I have converted three guns.
A Sig P224 originally in .40S&W. I dropped in a factory .357Sig barrel.
A G23 gen 4, using a KKM barrel.
A G22 gen 4, also using a KKM barrel.
I also purchased a G33 Gen 4, which is just a G27 with a .357Sig barrel. But do have a KKM .40S&W conversion barrel for it.
Nothing else about the guns needed to be changed. .40S&W magazines feed .357Sig just fine in my experience. Though the reverse is not always true.
The additional pressure going from 35,000 PSI in the .40S&W, to 40,000 PSI in the .357Sig, is contained by the chamber and barrel. And to a smaller degree by the lockup of the barrel and slide. 9mm +P is rated at 38,500 PSI, so if the gun is designed to handle 9mm +P and .40S&W, it can probably handle the additional 1,500 PSI of the .357Sig.
The other issue could be slide velocity. A lighter bullet being pushed faster can cause higher slide velocity. Is that an issue? Maybe, or maybe not. Run some data through this calculator to compare the two cartridges.
https://shooterscalculator.com/recoil-calculator.php
My .357Sig range load is a 124gr FMJ TCFP running at 1,425fps from that converted G23. I probably have a couple thousand of those through the gun now, with zero problems so far. The G22 probably has close to the same. That's not a huge volume of ammunition, so longevity is still a question, but who says you have to run full tilt ammo. You could run 125gr at 1350fps (SD factory levels) and have no issues at all.
I've got no personal experience with the BHP, but it must be a modern firearm. I can't see any reason it wouldn't work safely, at least in relatively small volumes of ammo. You might install a new recoil spring, and track your round count. This is what I've begun to do with the Glocks. I plan to switch the RSA at 3K just to be safe.