You can't have your cake and eat it too..... if you're looking for a lightweight rifle, you are going to be saving weight at the expense of barrel contour. So a lightweight barrel is going to be thinner; therefore it heats up faster. You can still get a lightweight barrel that is accurate, but don't expect it to shoot consecutive 5-shot groups and not see some thermal drift.
You mentioned this will be your first rifle, what makes you think you think you need one that weighs 6 pounds? It might help you get some better suggestions if you share what you plan to do with this rifle. It sounds like you might be suffering from a bit of misdirection, and don't know what you don't know. From my past experiences, a 6 pound rifle will be very easy to carry, but not very fun to do anything else with. Ultralights are harder to shoot accurately, and the weight savings are going to come at the expense of heavier recoil and not very much fun at the range. Ultralights make sense if you're backpack hunting in the mountains, and you carry a lot and shoot a little.
For everything else, a medium weight rifle make a whole lot more sense. Get a Tikka CTR with a 20" barrel. It's a really great starting point. Save the other $1500 in your budget, and go shoot the shit out of that CTR and get some experience. Once you know how it all works, you can make an informed decision on what you need and don't need. Right now, I get the impression you are going to waste quite a bit of money on something you may or may not need.