A London Terminals ticket isn't valid on the tube.
However the Oyster/Contactless fare is only £2.40.
You mention frequently, I suppose if you meant most days you would have likely stated that. For that reason I’d stick with a London Terminals season (if seasons are the way they want to go) and use Oyster or Contactless for the underground journey as others have pointed out.
OT, it’s an excellent hospital and they’ll be looked after well, though you probably know that already.
I doubt that would be any cheaper tbhHas he priced up renting a room in SE1, SE5, SE17 as an alternative?