I have an LMT standard 16" carbine. It costs around $1000-1100. Not $1700. Now, LMT's "Monolithic Rail Platform" rifle costs a pretty penny more, but the standard carbine is around the $1000 mark. The LMT rifles are very good rifles, though, I would not quite say they are better than Colt. Rather, I consider them equal.
S&W M&Ps can vary in price depending on what's on them.
The Colt LE6920 carbine can be had for around $1000-1100 at this time.
You are on the right track with these brands, just shop around for a better deal and one configured to where you want it. Buy what you want of those three brands, for they are all good. Other brands to consider that have comparable quality include Daniel Defense, Bravo Company, Charles Daly Defense, Noveske, depending on what is and is not available in your state.
There's a small gap between S&W and Colt/LMT quality. Buy what makes you happy, and if the S&W M&P-15 FT is what you are looking for, it looks to be a good rifle. . There is a larger gap between Colt/LMT and lower quality brands.
Now, I would NOT recommend buying a rifle with an A-1 or A-2 style fixed carry handle (which it sounds like that Colt AR-15A2 has). Removable carry handle uppers (flattops) are much better suited to optics. Even if you eschew optics at this point in time, you will probably want to try them later.