Yes, but you will have to go through T5 security, take the train to T5C and then back to T5B (or maybe T5A - not sure if buses still pick up at B), take the flight connections bus to T3, go through T3 security, use the T3 lounges, find your way to the arrivals stream of T3 (possibly requiring a security escort?), take the flight connections bus to T5, find a BA agent to reset your Fly status as your boarding pass will not let you back into T5 since you have already entered, then finally go through T5 security *again*.

Or maybe since your ticket have free same day changes you'd rather just get to your domestic destination earlier and not bother with all this faff.

Now if you didn't have a checked bag, it wouldn't be so difficult as you could take the flight connections bus from T4 to T3 and use the lounges (assuming you have your BA boarding pass), then when done you can go to T5 and you won't have the Fly issue. You would only need to do security twice rather than 3 times.