A Few Questions.

I might be wrong, but here goes.

38+P is still pretty low pressured. I would not be worried about loading them in modern 38 cases.

Brass is the same just head stamp is different.
Load data should be good just start low and work up.

Like the others said, the brass is the same with a different stamp.
TMJ bullets are plated. They can be loaded using high end cast data to low end jacketed. Since you are loading to 38+p in a 357, I would just use the xtp data.

Your saying all your handguns are 357 mag. Not only can you load 38 spl to 38 spl +p pressures you can load them to .357 Magnum, at least mid range magnum loads.
The cylinder in the revolver holds the pressure, not the case.

.357 mag brass is thicker but the extra thickness will not make a hill of beans difference with middle of the road .357 magnum pressures as long as the cylinder of your revolver can handle the extra pressure. Any .357 mag revolver can handle the pressure of any .38 spl load you can find data for with 38 spl cases.

Look at the 38-44 loads from about a 100 years ago, they were 38 special cases (probably baloon head cases) loaded to somewhere around midrange .357 magnum loads relative to the 1935 .357 magnum which was a lot hotter than most .357 magnum loads are today.

And that was with their present day steel.
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Now a 38-44 was a hot loaded 38spl shot out of a Smith and Wesson 44 special sized gun (N-frame). It had a lot of extra steel in the cylinder at the thin spots.

All I'm saying is the cylinder of the revolver is what handles the pressure, not the brass.

All 38spl brass is pretty much the same except for some of the wadcutter brass that stayed thin further down the case because of how deep you had to seat the bullet.
Just because the case says 38+P on it, doesn't mean it will handle more pressure.

It is to identify the load that someone put it it, so it doesn't get fired from a pistol with a cylinder that won't handle that kind of pressure.

CAN I USE THAT POWDER REQUIREMENT DATA ON MY 125 TMJ BULLETS?

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As long as it is a jacketed bullet and not a plated one.
Some bullet manufactures make plated bullets that are, quote, "Thick plated bullets". The plating on the bullets is thick enough, and the lead core hardness is thick enough to stand up to 1500 fps.
And they will without a doubt. These are fine.
But, the plating material is not near as hard as the alloy that a real bullet jacket is made out of. They will take the extra velocity, believe me I know, but they will copper foul your barrel faster than Jacket material will, when the jacket material is at a 32 to 35 brinell hardness compared to the hardness of the electroplating of copper on the lead core of plated bullets that has been simply put on thicker.

The difference is in how much you shoot them. If you only shoot a hundred a month you may not notice a difference. So take my advise with a grain of salt.
I shoot .357 magnums 50-100 a week and I have noticed the difference in copper fouling.

edit: The difference is not alot and most people won't notice it.
Shoot away, your good with what your doing.