Not necessarily arguing, but I've never heard of a TSS load damaging any barrel, .410 or otherwise.
Shooting TSS through a .410 would be much better than shooting steel. Full disclosure, I started hand loading before TSS was available commercially, and I haven't tried to keep up with all the commercial options. But unless the manufacturer is an idiot, TSS loads all use a thick, heavy-duty wad designed for hard pellets, to prevent scoring. And commercial loads—lead, steel, or tungsten—SHOULD all be within SAAMI pressure specifications, so chamber pressure usually isn't the issue. But when considering the potential risk to a barrel, the biggest distinction between steel and TSS is the tiny, smooth, perfectly spherical TSS pellets. Because steel (< 8g/cm3) is comparatively much lighter and less dense than tungsten (>18g/cm3), you have to use much larger shot with steel i.e., 2 or 3 shot versus 9 or 10 shot. A shot column with tiny, smooth pellets is not likely to bulge a barrel like steel can.
I'm over simplifying it a little, but this analogy helps me conceptualize it. Picture a 2" pipe that narrows slightly at one end. Take 8 oz of marbles, tie them very tightly into a sock, and try to ramrod that sock from the wide end of the pipe through narrow end. Then do the same thing with 8 oz of BBs. Which sock is more likely to get lodged in the pipe when it narrows?