In auto's the bullet profile dictates the seating depth. I too load Precision 125's, they have to be seated shorter than what I like but they function fine. I was loading another brand out to 1.125 which gave 100 percent reliability. I think the Precisions are at 1.10 in my gun presently. These are both TC profile bullets. The difference in my two bullets is the shape of the cone. The precision bullet has a "fatter" nose, whereas the other brand the nose would fit into the barrel without contacting the rifling. It just so happened that the amount of bullet in the case is the same, the Precision bullet is shorter. This is a very important point when reloading. If you seat a bullet deeper than before, using the same charge you will increase pressure. This can be dangerous at the worst but will also affect accuracy.
You might try some other bullets before reaming the chamber, see if you can find one you like that works the way you want it too. You could also look into a different barrel with more generous dimensions, then you would still have the stock barrel. Just a thought, hope you find a good solution, let us know.
Best, Rick