The correct answer is whichever bullet shoots well. More specifically, that will largely depend on how long the chamber/throat of your rifle is, and how long (heavy) of a bullet can you load optimally for it. Without knowing that information, any answer you get here about a specific bullet is pure speculation. In general, you'll likely have the best luck loading the longest/heaviest/highest BC bullet you can load optimally in your rifle. That means not sunk way down in the case, or just barely seated in the neck for starters. Add in that within that range, you want to be able to achieve the seating depth where a particular bullet shoots the best. Selecting based on these parameters will help to ensure that you can develop a load with optimal powder burn and get the best velocity [and more importantly, precision] possible out of the relatively short barrel.
So, if you're rifle has a fairly short throat, 155s, or possibly something like the Berger 168 Hybrid or 175 OTM Tactical bullet. Those have sufficiently short bearing surfaces that you don't need a really long freebore to seat them optimally. Possible powders would include Varget, H4895, possibly 8208 XBR. Something a little on the fast side will help get the lighter pills moving in a 20" pipe.
If your rifle has a very generous throat, bullets in the 185-200 gr range would be a possibility. 185 Juggernauts, 185 Hybrids, 200 Hybrids, possibly even 208 A-Max, although those have a very long bearing surface and would require a pretty long throat. Based on the limited info you provided, I'd guess something in the 168-185 gr range might be your best bet, but again that is just a guess and totally depends on your throat.
The bottom line is that you don't need ridiculous velocity to shoot a .308 bullet out to 500 yd, so precision and ease of load development are what I'd be aiming for, regardless of the bullet you choose. The rifle will tell you what the optimal velocity is out of the 20" barrel for a given bullet. In addition, you can run the numbers yourself at JBM Ballistics (or similar), but you will generally find that the longer/heavier bullet with a significant BC advantage will win out over a shorter/lighter lower BC bullet, even though you can push those faster. Unless the BCs are fairly close, it's typically difficult to push the lighter bullets fast enough to overcome the BC deficit.
FWIW - I have a DTA SRS Covert .308 with a 16" 8-twist barrel that I've shot many different factory loads out to 600 yd with no trouble. MVs ranged from ~2500-ish (175s) to mid 2500s (168s), to ~2800 fps (155s). Although clearly not as good as my handloads out of a 30" F-TR rifle, the precision was pretty decent with a few different factory ammos, so you ought to be fine at that distance with your 20" barrel.
