You've got a slight problem.... Bullets that work wonderfully in faster cartridges aren't going to behave the same way out of your little 308. You've got an already modest cartridge, and its been neutered by that 16.5 inch barrel....
I've used a 20" Remington Model Seven on deer and caribou for about 25 years. It works. However, I have come to realize that velocity losses due to the 20" barrel mean I need to be careful with bullet selection. this will be even more important for your 16.5 incher....
Hard, deep penetrators like the Barnes ttsx are great bullets. However, they really need speed to open well. they work wonders in various 300 magnums and similar 3000 fps+ cartridges. I suspect you'll likely have a hard time topping 2500 fps with a 165 ttsx out of your gun. If this is the case (and its likely) you'll find those fantastic TTSX bullets simply aren't going fast enough at 150, 200, 250 yards to really open up and act like they usually do. Expect miserable, disappointing results. Its not the bullet. Its a case of inappropriate use of the bullet.
Given your initial muzzle speeds (due to barrel length), you need something that will open better at reduced speeds. I've have decent luck with Nosler Partitions. Fortunately, you've got new developments in bullets, thanks to the long range shooting crowd. The problem with long range impact is, you guessed it, reduced impact speeds. Sounds familiar? The Nosler Accubond LR and the Hornady ELDX are designed to expand well at reduced impact speeds, which is exactly what you need.
Don't worry about bullets "breaking up". Yes, old school cup and core bullet would break up at high impact speeds and close ranges. Your 308 simply won't drive one fast enough to be a real issue.
I'd grab a 165-168 weight, in one of the long range variants, and call it good. Remember... since you've got a 16.5" barrel, you are already at a SIGNIFICANT velocity disadvantage. While many 165 grain 308 loads are rated at 2700 fps, many will only get an actual 2650 or so out of a 24" barrel. I'm getting low 2600's/high 2500's out of a 20 inch. You'll likely get something closer to 2450-2475. In other words, your initial muzzle velocities at powder burn distances are already reduced to what impact speeds would be out of a 24" 308 at 150 yards or so. Or 200 yards for the same bullet out of a 30-06. In plain english, when you sight in on that bull at 250 yards with your little rifle, the bullet will act much like the same bullet would out of an 30-06 would at 400 yards.... Odds are, even if you can reliably high the target at 300 yards with your rifle, you simply don't have enough real velocity left for the bullet to open hard, expand, penetrate and get it done reliably.
Hard bullets won't work well at any reasonable range. You cannot push them fast enough. Opt for something softer, and easier opening. the ELD X and Accubond LR should be ideal.... they are designed to expand at reduced speeds....
for what its worth, I've been using 165 grain Ballistic tips out of my 308 for most of those 25 years. Yup. they are a fast and fragile bullet. I would not dream of using one in a fast gun. However, at the modest short-barrelled 308 speeds of my model Seven, they work WONDERFULLY. Absolute hammers on deer and caribou. They work MUCH faster and more reliable than other bullets of harder, more controlled expansion. The reason? My 308 isn't fast enough to really work the magic with a hard, deep penetrator but those Ballistic Tips are damned near ideal at the 2500-2450 fps impact speeds I usually end up shooting my game at.
Match the bullet to your impact speed and it all works well