The generally accepted brands around here are Colt, Armalite, Bushmaster, Rock River Arms, and DPMS.
I know you would not be disappointed with any of them
You will have little problem getting an M4 style AR15 for that price.
Since I am familiar (and very satisfied) with RRA, I will point you in that direction.
Around here, ADCO firearms is the place to buy RRA equip.
Here is a link to their
RRA rifles page...You can save some money by purchasing a complete upper, and a complete lower, which is how many buy their AR's. Check out the "Complete Uppers" section, under "Rock River" on the left menu, along with the plain "Rock River" section for lowers. Notice there are collapsible stocks available. Steve could swap and A2 for a collapsible CAR stock, or set it up however you need. He's very, very patient, and helpful when you get started.
Here is an example:
Tactical CAR A4
16" 1:9 barrel M4 handguards
$816.67
M4 ENTRY
chrome lined barrel
$854.44
Or.....
Lower-Complete
standard trigger
$215.56
plus one of these complete uppers:
M4 ENTRY
chrome lined
$566.67
LETactical
LE Tactical CAR A4 Upper M4 handguards
$532.22
A3 carry handles are listed with the "Rock River" parts section and are $110.
I would suggest you get an LMT back up sight, rather than a carry handle, unless you are really set on one. (ADCO carries those, too -
look under "LMT" section). I have one mounted on my RRA M4 style.
Here is a pic of my setup, similar to your intended rifle. I went this route becasue I want an AIMPOINT M2 red dot scope. This would still fall under your $1400 total. Same price for LMT as A3 handle, but leaves ome space on your upper for optics, if you desire. NO ONE carries AR's by the carry handle. It would make you point it at people.
If you go this route, you simply need to buy a magazine, pin the upper to the lower (the pins are part of the lower, you just push them together to lock the two pieces) and shoot!
If you do decide to build it yourself, I'd be happy to help you through it, as I just did it last week.
I even have a spreadsheet detailing costs and parts and tools for the rifle pictured in my link.