This has not been thrown out yet and would be an option if you are looking for accuracy - a stainless steel barrel.
An SS barrel will not be as durable as a chrome-lined barrel, but they are generally more accurate. You could do a few things:
1. Buy a BCM lower and a BCM SS410 stainless upper. You could come in at the higher end of your budget with a very accurate and high quality rifle.
2. The Larue predatAR is an AR with a thinner profile SS barrel. It would run higher than your budget but it is a very fine rifle and should get you 1.5moa groups and smaller. Some report sub-moa groups if the barrel isn't allowed to heat up.
3. Buy a complete lower receiver and have Rainier arms build you an upper. They have a lot of high quality components and many, many folks say their customer service is second to none.
I would recommend that if you go with a SS barrel that you go either 1:7 or 1:8 with a Wylde chamber or comparable chamber. They will shoot both 5.56mm and .223 rem. A barrel with a 1:8 twist should give you the ability to shoot up to ~77 grains. It will also shoot 55 grain bullets just fine. In fact, a 1:7 barrel should shoot 55 grain bullets just fine, IMHO.
With the right SS barrel you wouldn't add a lot of weight and would have one heck of a shooter for an semi-automatic rifle. But, it is not optimal for mag dump sessions. A chrome-lined barrel will hold up longer.