Well...all of the ammo you have listed is hardly "match" quality or otherwise capable of achieving precision levels of accuracy. The reported average of 1.5MOA that you are experiencing is par for the course with the ammo listed out of your setup. Similarly, Stag's basic AR barrels in my experience are about a half-a-heartbeat away from being glorified tomato stakes in terms of their ability to produce precision levels of accuracy (which it sounds like that's what you are after) and a 1:9 twist, while it'll work ok for 55gr ball and bullet weights in/around that range, I would move up to a slightly faster twist (1:8 or faster) and I would run heavier, match quality ammo/reloads 69gr to 77gr pills of your choice (which should not only help with accuracy, but also with performance at extended ranges).
Before I would go upgrading to a new barrel, I would first go ahead and get rid of the stock trigger and move to the Timney you mentioned (if you want a single-stage) or a Geissele (for a 2-stage). That should be a BIG improvement in and of itself. A quality optic and mount and/or rings is also a worthwhile investment that you'll be able to use with the rifle regardless of the barrel you have on it. I cannot stress the importance of a quality mount or rings when putting any optic on an AR, as well as properly mounting said optic. Also, get yourself some quality factory ammo that is proven capable of producing the highest levels of accuracy across a wide range of platforms (i.e. - Black Hills, FGGM, SW Ammo, Corbon, etc.). Once you have all of the pieces put together and you can dedicate some quality range time to testing your rifle, then you can make the call as to whether you want/need to rebarrel based on your results.
If you end up going down the rebarreling road, then there are a host of different barrel options depending on what you ultimately want/need and how much you are willing to spend. You'll also need to factor in, if you don't plan on doing the work yourself, how much it will cost to get your upper into the hands of a qualified smith to have the barrel swapped out. If you aren't doing the work/don't have the necessary tools for the job, you can get a smith like Steve and the guys from ADCO to do a simple disassemble/reassemble with a new barrel job for ~$100 including shipping both ways.