You have not state if you are looking for non lead ammo or not? You also did not state what type of deer you expect to shoot, both can impact recommendations.
I've shot everything from 100 grain to 165 grain in my .308's for deer, I also reloaded most of my ammunition. Most of my rifles would not shoot heavier bullets (180gn plus) very well due to the twist of the rifling.
A 20" .308 barrel will shoot just about any reasonable ammo at a good velocity, don't worry about your barrel length. I shot a 20" .308 and killed hundred's of deer, some at great ranges. If I was shooting lead ammo I would pick a lower cost 150 grain ammo as this is typically the most common ammunition weight available and can be purchased at a reasonable cost. If shooting non lead a slightly lighter bullet works well, maybe in the 120 - 130 grain range. This is because mono bullets work well at a little higher velocity. Top of my list would be Barnes TTSX in 130 gn but Sako Gameheads 123gr are also a good selection.
Once you have found a round that your rifle shoots well stick with it for deer stalking and not spend time thinking about your ammunition. Consistency is your ammo is key.
As a reloader I experimented a lot on using lighter ammunition for smaller deer, typically 110gn in .308, I loaded them to the same velocity as 150gn ammo and they worked well.
Good luck with your stalking.