This isn't going to be what you want to hear, but your choices for fully assembled A1/A2 uppers is going to be extremely slim pickings right now. Even companies who offer them for sale still (J&T, Del-ton, DPMS, Bushmaster) are all out of stock of A2 receivers to build uppers with and no ETA as to when they'll get more (trust me, i've tried them all when I was building an A2 Carbine a month or so ago).
If you must go with a fully assembled one, the EE is going to be your best bet. I usually see several 16" A2 uppers for sale each week. They usually go quick, and aren't super cheap, but most for sale seem to be in good condition. If you're diligent about scouring the EE, you will find one eventually.
Option 2 is just building one yourself. I know you don't want to invest in the tooling, but chances are you will build more ARs down the road, and those tools will be a godsend. All you need is an Armorer's wrench, vice blocks, a torque wrench (I use a $25 Craftsman beam-style torque wrench, works fine for the purpose), and possibly a breaker bar. The cost of buying the tools will only be slightly more than paying someone to do it for you.
Building one yourself gives you far more flexibility. Stripped A2 uppers are plentiful on the EE and gunbroker. Del-ton will sell you an excellent, chrome-lined barrel kit with the proper non-F marked FSB for a little over $200. If you want to go even higher-end for your barrel, you can get a stripped barrel from BCM and have ADCO pin a non-F FSB for you (Del-ton sells stripped non-F FSBs or they are common on the EE).