Can a Godox X1T remotely fire my camera shutter and trigger an AD200 at the same time?

No, this setup won't work. You're close, but not quite right. The X1T is not a receiver (hence, the "T" suffix) or transceiver — it is a transmitter only. So one X1T cannot trigger another X1T.

You still need all the pieces you mentioned, but you also need an X1R to trigger the camera shutter. With wireless remote shutter and flash, you have two events:

1. The first event is the wireless signal to trigger the shutter. This is accomplish with an X1T + X1R pair (transmitter + receiver). The X1T is in hand, the X1R is cabled to the camera's shutter release port. Do not put the X1R into the camera's hotshoe.

2. The second event is the camera signaling the hotshoe flash to fire. This triggers the second X1T to pop the AD200 (i.e., the 2nd receiver).

Make sure the X1T+X1R pair are using a different channel than the X1T+AD200 pair.

(See also: Godox - operate camera and speedlight remotely discussion at dpreview.com)

It does seem sort of silly to have two X1's attached to the camera (one attached only by shutter release cable, the other mounted in the hotshoe), but it's really not different than using a X1T+AD200 in conjunction with a different brand dedicated wireless shutter release (such as the Vello Shutterboss, or Phottix Aion, etc.). In fact, personally, I'd probably go that route (if only because dedicted remote shutter releases can also have timers, delay timers, sequences, etc., that you can't do with a simple X1T flash/shutter trigger remote.