Several factors in play here. None in support of your proposal.
1. Steel shot ( actually unhardened iron) is harder than composite ( damascus, twist, etc) barrel material. Much harder. If it compresses or scrubs through the plastic shotcup on it’s way down the barrel it WILL mark the inside of the barrel. In many cases it will even mark modern steel barrels and did so with some of the earlier steel shot ammo until new stiffer and thicker wads were designed to overcome this. It’s not uncommon today to find shotguns with steel shot damage from the 1960’s, ‘70’s.
2. Steel shot does not compress. Unlike much softer lead, when a closely packed group of steel shot slams into the beginning of the choke constriction it won’t compress to squeeze through. There is no give, no way for that group of shot to get through except by momentarily expanding ( opening) the choke diameter to let it through. This can and frequently does cause a permanently bulged barrel just before the choke even on the best of tightly choked modern steel barrels. Composite steel barrels deform even more from this abuse and can also crack. If the barrel has little or no constriction ( choke) then the steel shot will pass without bulging the barrel.
3. Another factor to consider is the tremendous extra strain you are putting on a gun that is likely well over 120 years old, maybe more than 150. Aside from the original integrity of the barrels ( and you have no way of knowing how well made they were ) you are putting a huge extra strain on the solder joints between the barrels, the ribs and the lump ( the part that locks the barrels into the action). That beautiful ancient walnut stock is now brittle and somewhat vulnerable to excessive shock, especially around the locks and at the grip. Your underlever action, if a typical Jones type double screw grip, is plenty strong enough if still well fitted but the extra recoil caused by the unyielding constriction can only cause negative results on a vintage gun.
By far the best solution is Bismuth shot. Very effective, readily available, ample loading data available and it won’t damage your gun. Upsize one size of shot ( use #4 or #3 instead of #5, #1 or BB instead of #2), stay within the range of your gun and you’ll be pleased with the results. The only negative is the higher cost of the Bismuth shot but for hunting only as opposed to target use your ammunition will be very minor compared to your gas cost to get to and from the hunt.