Here are the numbers from the barrel starting from the barrel end back towards the receiver:
758
M.L.MS (the L could be and I)
r.R (the r could be an L or I)
15.9
.94
1.6.31
H.M. Pope
*The font that the stamps are in can be misleading into what actual letter they are, indicated by the notes next to each stamp.

Some guesses on my part.

758 is the Jersey City barrel number.

M.L.MS is a mystery to me.  Could it be the type of steel Pope used for the barrel?  Can some of you metallurgy guys help out?

L.R. (?) Did Pope intend this barrel for Long Rifle bullets only and not as a varmint rifle?  The date makes me think that.

15.9 is the twist; 1 turn in 15.9 inches.

0.94 ?  Does this indicate a slight gain twist? Or maybe that there is a choke in the barrel?   

The Pope barrel I am currently shooting is a Short, 25.0" uniform twist with a "0" stamped below it.  There is no choke in the barrel.

1.6.31 is the date, 1931.

Gentlemen, after much research and history of H.M. Pope and his methods of making barrels the following information about this rifle is what I have discovered.
It is a Model 12 reworked by H.M. Pope. My grandfather must have commissioned Pope to put a match barrel on the rifle. The understanding is that in order for Pope to make a barrel, you had to take your rifle to his Jersey City shop. Here you would leave it with him for up to a year in order for Pope to make the perfect matching barrel to the type of rifle, it is also understood that Pope modified the rifles to be match ready.
Knowing that my grandfather competed in multiple small bore and large bore competitions in the New York area including many trips to Camp Perry matches (have a box full of medals from these matches). Knowing the collection that our family has and the history of my grandfather, he had Pope setup this rifle for his competition matches. These Jersey City barrels were never sold separately, you had to have it commissioned for Pope to make. Also, considering my grandfather was born in 1909 and started match shooting in his teens makes the timing of the barrel stamp of 1931 and the age of the Model 12. This makes perfect sense to show the barrel was not added after Pope's death in the 1950's.

If you look at the M under the barrel number and the M in Pope's name and then the one to the right of it it is pretty obvious they are not the same and added at a later point in time. Possibly by Pope, previous owner or possibly someone else. The quality of the name stamp is also a bit suspicious as are the double dots H:M:Pope especially the two dots located higher after the letter H and unlike any others I have seen and it does not appear to have been done by his hands or by using the same stamp set. 

JLouis

Enlarge it and look at it closely it has no resemblance at all to the other letter stamps that were used and you are correct it does appear to be a double stamp but still doubt full Harry did the stamping and even more so it being done at the same time when enlarged and then doing a close inspection. You will also notice non of the letters are in straight line and not at all typical of a one piece name punch and even the metal displaced from the punch left raised and not typical of Harry's work. OleTuck/ Fitz has probably seen more original stamped Pope barrels than most and hopefully he will chime in with his thoughts and prove me wrong should that end up being the case. 

JLouis

Not insisting on anything everything else stamped on the barrel looks correct just pointing out the name under the barrel being a bit different and possibly added later. The OP was asking for opionions and I was giving him mine on the under barrel name stamp only and I never said anything was fake or the barrel the rest of it all looks proper to me. 
It's a very stunning and beautiful rifle and the history quite rmarkable and even more so by remaining.in the family. 

JLouis

Bill have a past ASRRA Journal that lists the known Pope barrel numbers at that time, what they are and what they were attached to and I will dig through my pile when time allows to see if the OPs might possibly be listed and my hopes as it would be real nice addition to what he has already uncovered in his research thus far. Unfortunately I can be of no help with your request for the information you are currently seeking.

JLouis