I’m pretty sure WW Airport is only open business hours, but it can’t hurt to try and call to let them know about a planned flight. I tried but every time I called the number just rang.
Also, I’m sure I checked Dronescene when I did fly and only recall the local airport, not the bigger military area being listed at that time. Anyway, Dronescene shows a number for the ATC so again, call to advise of intentions.
Once done, take off from pretty much anywhere on public land - just be sure to have the permissions in place (even if only verbal) so when (if) Mrs Busybody from the castle comes out you can answer. See here: CADW - no drones policy - #8 by Jhdee and here: Kidwelly Castle Challenge for more on that.
Have fun.