I've loaded with Federal's 129 grain Plus P Hydra-Shok, and the factory PR man had it chronographed for me by an engineer. (I'm a gun writer and this was for an article. He was very helpful.)
In a M-36 snub, it clocked 836 FPS, and from a three-inch barrelled M-36, well over 900 FPS.
The Speer NON short bbl. 125 grain JHP .38 round has clocked well over 900 FPS in three-inch guns, and I load this in my M-60-4, although I have a box of the 135 grain Short Bbl., and may switch to that. But the first is so accurate in this gun that I'm impressed to the point that I haven't changed yet. Home ddefense aside, I want to be able to hit a snake in the head at danger ranges.
I have also relied on the lead 158 grain Plus P HP ammo from Winchester, Federal, and Remington. Be aware that there is some Federal 38G ammo floating around that wasn't properly crimped. The normal version is okay and very accurate in my guns. I especially like this in longer barrels.
The lead load was issued by Dallas, Miami, Chicago, the FBI, etc. and established a very good record on the street, but usually in four-inch barrels. Still, there were some snubby shootings where it did very well. My brother found it to put down a large dog with a shoulder shot from a M-60 snub, but the angle of the shot didn't get it into the vital organs. The bullet was fired from above, down into a shoulder as the dog lunged for him. It did drop, then ran back to its owner's home. I'm sorry to say that it lived. Had he put that bullet into the body lengthwise or crosswise, into the heart, etc., I think we'd have one less dog problem. Still, a snub .38 is not the optimum answer for dealing with large dogs.
I talked with a Speer rep who told me that NYCPD issues the Short Barrel 135 grain round in ALL their remaining .38's and that it has performed extremely well. But I don't know how many shootings are involved.
You might want to check the thread here on this type of load and read Wayne Dobbs's comments. He was a detective with a suburban dept. near Dallas and got to hear comments on how local shootings went. Most involved four-inch barrels, of course. He posted that the lead HP round performed very well, although it was eclipsed by the 145 grain .357 Silvertiip, which he said did EXTREMELY well. I believe that was how he phrased it. But you don't want to be firing that .357 ammo indoors, for hearing reasons and because it might endanger persons in the next room.
I think you can choose any of these loads and if you place your shots well, you'll probably stop your intruder.
I do not use lighter bullets than 125 grains in .38 or ,357, as i want ample penetration. But the old Treasury Plus P Plus 110 grain round worked, too, from feedback that I've seen. However, light bullets may not penetrate well, and they don't shoot to the sights, if your gun has fixed sights or is dialed-in for nomal ammo for other purposes.
Your three-inch gun performs more like a four-inch than like a snub, so you have an ammo edge over a snub.
Pressed for a single answer, I'd tell you to choose the Speer Short Barrel or their 125 grain normal Plus P and see which you shoot best and to use that. Remember, placement is more important than the exact bullet used. But I think the track record of those rounds gives confidence. It is hard to get reports of actual shootings unless one is in LE, but those have enough background with large agencies that I trust them.