TTT1 is Namco System 12 and therefor JVS and JammaProbably the undamned usb decoders.
The game doesn’t matter exactly, but your setup especially for I/o, does. If you use, for example, a namco or capcom jvs io with the namco256 for tekken5, and are connecting that jvs io to a supergun that uses db15 and compatible pinout (like a HAS) then yes you can plug your controller into the undamned usb decoder, and then into the supergun.
TTT 1 or 2 I dunno, you could probably do the same thing but I’m not sure which io they support. Probably the same answer though.