It's not so much the speed (analog TV is only ~MHz BW) as the storage. A frame is, what, a few hundred kB? You need to store at least half that (one full field), plus an extra one half line. Play back the first line of the first field, then the just-recorded line of the current field (which will have finished receiving just in time to transmit; more likely, you'll store the whole line and not mind the extra half-line requirement), then the second line, then the just-recorded second line, then... Obviously, alternating down as you go, at double the output bandwidth.
Mid-tier ARM MCUs have about as much RAM onboard, give or take exact bandwidth, bit depth and encoding. And they run in the 160MHz range which is enough for a dozen or so instructions per sample, which may facilitate some minor encoding to save on space. Figure about $10 for the chip, and, who knows, several months to write it..?
Might be easier to write on an FPGA, a mid-tier one of which might run $20-40, would have enough block RAM onboard to hold a frame or so, and way more than enough logic to handle any decoding and encoding you might need to support the video format and to save RAM.
The PC can't just use a capture card?
Tim