If the pistol has the integral rib it is a MkII. The MkIII pistols do not have this rib. This rib is on the exterior top of the slide. I think we may be speaking of different things so I've included a couple of pictures.
All of the Browning HP bbls that I'm aware of are "lugged." The picture below is of a used Barsto HP 9mm bbl I own. You can see the two lugs that lock into the slide in front of the chamber area.
This next picture is simply a close-up of the same bbl. You can see the lugs a bit more clearly in front of the chamber as well as the protrusion I spoke of at the rear of the bbl above the chamber entrance right above the feed ramp.
The next picture is of a 9mm MkIII. The top of the slide has no rib, just the fixed sights as shown. Now, this gun does not have the right-side ambidextrous thumb safety lever show, but that's only because I removed it. Stocks are not factory and the hammer's been bobbed and the gun refinished, but this is about as close to stock as I have. The important thing is that you can see if the top of the slide looks like your gun or not. If yours does not have the rib, it is not a MkII. I think it likely is a MkII due to the serial number and the MkIII pistols had not come out if the second letter is a "V." My earliest MkIII has the serial number, "245PNXXXX" and was purchased in '89.
I think that the MkII began production in '84. Mine had the "245PV" prefix and I bought it in '85.
...and Barsto makes a very fine bbl in my opinion.
Best.