Once you’ve got your bags you need to go to T5. You cannot check in / bag drop for a T5 flight at T3.

So you’d need to clear security at T5 to then go airside to T3 (you can’t enter T3 by normal security without a T3 BP) then clear connections security at T3.

Then on return to T5 you’d clear security again,

there are several posts on this topic on the BA board. It’s really not recommended.