Hey Baker42, I'll tell you what my tactical beard tells me after spending lots of dough going down similar paths.
-First off if you're planning to compete and unsure of what dicipline the .308 will leave the most doors open.
-sounds like you shoot well and have a stock rifle that shoots well, however your not likely going to get huge gains in accuracy no matter what you do to it in its stock form (already mentioned).
-spending $2-2500 will get the rifle the way you want with top shelf components using your current barelled action and likely be a .5moa shooter....keeping in mind the further out you push it any minor tolerence issues will be exploited...but will be a very nice rifle.
If you have some coin now...will you have it again later if you want another rifle? Something to consider..
My first adventure similar to this was taking one of the very early 5R .308 and sent it out to ATRS and had a full option A5 installed and bedded, Jewel trigger, muzzle brake, bolt knob, and refinish with cerakote...this pretty much ate up a similar budget but was a very nice rifle....and shot marginally better (honest .5 moa). Buddy of mine shot it for a day and wanted it so I sold it to him.
Since then I tried different chassis models, more McMillans, HS Precisions....had more rifles (full customs built), and was never totally happy....but spent lots of $$ figuring out what I want....an A5 in .308...LOL!
My buddy still has that rifle, it has many thousand of rounds through it, all beat up, the jewelled bolt pattern is long gone, and it still shoots .5 moa and he can get a first round hit at 1000 yrd gong still! Throw a new barrel on it and it will keep going.
Lots of money later....I have a couple of my favourite .308's left that I won't sell...My interest now is in .338 Lapua and I've been bitten by the pesky BMG bug. Had I been more patient I would likely still have that first 5R and still be wearing it out along with some new ones too

That being said, you can customize your Remmy and get it just the way you want and shoot it and compete with it. Once you compete, I'm sure you'll find you will want more out of the rifle and can always do that later.....a trued action and good match barrel will get it in the hunt at any competiton, but the stock barrelled action won't hang with the F-Class guys, but will be good enough to try it...as an example...