I bought an Israeli surplus MK3, swapped out the springs, sent about a case of cheap ball through it, and now I'm ready to carry it.

All of the 9mm carry ammo I have on hand is the +p+ Winchester Ranger TA.  Will a BHP handle that?

If not, are they okay with +p? GDHP 124gr +p is the easiest for me to lay hands on besides what I already have.

I did some research when I got my Inglis and I think it was generally not a good idea.

I would think a stronger hammer and recoil spring might be the answer. Contact BHSS, they will give you guidance.
https://bhspringsolutions.com/

I vote no.
Mine love 147gr standard ammo

The BHP was made before the advent of +P ammo and most Hi Power experts will not recommend running hotter ammo in a older BHP. While the MARK III cast frame pistols do hold up better for those shooting large amounts of hot-loaded 9mm in either handloaded form or +P, they will wear it out sooner than other modern pistols with hot ammo.  Also, it has a weak barrel lug prone to shearing which is why the guns have a 2-piece barrel. It is actually a strengthening measure to use a steel with opposing grain structure from the tube. This is why Bar-Sto aftermarket barrels are also two piece.

ETA:  There is no SAAMI spec for +P+ ammo above the already over-loaded +P designation.  Since the load is supposed to be still below the pressure of the proof cartridge, it will work  if shot rarely in your BHP. But prolonged usage will wear out your gun faster.  Because of this fact and the abomination of a trigger linkage on the BHP,  I sold my BHP and went to JMB's other design, the 1911 in 9mm. The 1911 in 9mm can handle the hot rod 9mm rounds without accelerated wear. Most 1911 ramped barrels offer plenty of chamber support for hot 9mm loads.

I have a Hi-Power clone and I just stick to 147 grain Gold dots.  The possible small loss in performance is worth it to know that I am not beating my gun up.

When I had my MK III BHP I ran lots of NATO, +P and even +P+ but I installed an 18lb spring and a buffer before doing so, for reasons.

Personally, I would not run anything but standard pressure 9mm in a forged frame Hi-Power, again, for reasons.

How is the BHP for running Winchester 9MM 124gr NATO loads?
I bought several cases of it because it does run my UZI SMG....

Fo or no fo?

I've run this ammo through my Hi-Powers for years with no issues. Use a standard 17lb recoil spring.

http://www.hipowersandhandguns.com/index.html

I ran a couple thousand rounds of +P through an old IDF gun many years ago with no effects.