Take your TV2 remote. If it came with the 622, it should be labeled "IR/UHF Pro 6.3" on the back. Note the word IR. Set the remote channel on it to the channel of TV1's IR remote, but without using the system info screen to change the receiver. It will then give off IR signals (not UHF, line of sight only) that can control TV1 and not TV2.
Thus it is easily established that the remote will in fact send IR signals, contradicting what was posted earlier in this thread.
Now, for the matter of sending IR signals to control TV2 from your media center (and not from your remote control, which is what I am inferring you are trying to do), it should work, but you'll have to have the correct IR codes for your media center to send. This means, among other things, the right codes recorded for the right remote control channel, namely the channel that is shown for TV2 on your system info screen.
So, set your "green 2" remote back to the channel for TV2 and double check it works for TV2 again, then use the learning function on your media center to get the correct IR remote codes for that remote channel.
YMMV, try it with one or two remote codes to make sure it's working before learning all the possible codes.
YMMV because I haven't confirmed that the 622 will LISTEN for IR codes from remote 2, just that remote 2 will send them, but it's worth a test.