If the revolver dates from 1936 it's more than safe for smokeless powder. The ammo...depends on when it was made. Smokeless powder and the .38 Special were developed at roughly the same era...more than likely it's smokeless.
Show us some photos of the ammo, preferably if it is in a factory box. It's probably not black powder era but we can tell from the packaging. Unless the rounds are handloads with black powder which is possible but not too likely.
If you want to shoot it, the best factory ammo will be standard velocity 158 grain lead round nose bullets. That is what the gun was designed for, pressures will be safe and it will shoot to the sights as that is what they were regulated to shoot with. You will also be pleased with the accuracy of 148 grain full wadcutter target ammo. It is very pleasant to shoot.
The last Colt double action black powder revolvers made were the New Army & Navy models made from 1889 to 1903.
The last couple of years these were made in .38 Special and for smokeless loads.
The new Colt Army Special of 1908 was specifically made from the get-go for smokeless ammo, and every Colt made thereafter were smokeless.
After 1908 the Colt Officer's Models were all built on the Army Special frame, and specifically made for smokeless loads.
A 1930's Officer's Model is perfectly safe with any .38 Special loads, although using +P loads will increase wear.
Shooting virtually any amount of .38 Special loads in your revolver will not cause any damage.
These Officer's Models were built for 158 grain loads as the standard, and were usually used with 148 grain lead "wadcutter" Target loads for best Match accuracy.
In short, buy any standard .38 Special ammo, shoot as much as you want and enjoy one of America's finest revolvers.
Regardless, that thing will shoot BP just as well as smokeless. You must be diligent with cleaning after you shoot BP, or risk the rapid onset of corrosion. Your revolver will shoot smokeless till the end of time, but clean it anyway when you're done for the day. At least punch it out with a solvent patch or two.
That is a nice old revolver.
I have a case of the "Treasury Load"...really great spec ammo but it should be restricted to .357 Magnum revolvers only. My snubby Python loves that load...kinda strange since the Python barrel twist is more for heavier bullets. I haven't shot my new 3" Python yet...may have to try that load in it.
My personal theory is if you want or feel that you need more whack, the sensible thing to do is get a bigger gun. Bigger bullet or higher velocity, or both, the choice is yours. This practical approach does two things. You don't shoot your gun loose or accidentally demolish it, and you get to take possession of more guns. Win, win.
What you're calling the "Treasury Load" went to other gov't agencies as well. Way back in the day I was a federal agent and we were required to carry an American made revolver, 5 or 6 shot, in .38 Special only. Then they issued us that 110gr. jacketed HP +P+ loads for duty carry. We qualified with 158gr. lead HP rounds and carried the hotter stuff. We did fire it in our model 60s, but I much preferred it in my 66 - it's pretty much just like a .357 but back in the 80s they didn't want the press that "one of those nasty .357 Magnums" got if you actually had to use it. I still have one box of that +P+ in original packaging. It has a very large "NOT FOR RETAIL SALE" across the front of it.