[Brightcolours]

Frankly even if I had mirrorless I would move to your combo. And I happen to own both 7Dmkii that is in the same category as your D500 and EOSRP
for birding it is surely 7Dmkii, it is by far superior
1) reach with 1.6 crop
2) viewfinder and batteries  with a MILC when you have the camera on using the viewfinder for a long time, it will drain battery and the worst if often goes to sleep mode, none of this with a DSLR you keep your eye n the viewfinder all the time you want waiting for the bird to be in ideal position
3) faster to find/lock autofocus  on the bird, while a mirrorless camera shows superiority in portraiture with eye autofocus since in portraiture you have all the time for locking  focus on your subject then tracking , in birding it's different, before tracking you have to lock autofocus on the subject and here mirrorless cameras struggle and this tiny amount of time might mean losing your shot.
4) screen refresh rate, whatever you do a DSLR will always  be faster here it is physics
5) general feeling and handling, probably because we have used DSLRs for a long time, but we feel better with them

Point 3 is not right. Most newer mirrorless cameras are faster with acquiring focus. Your RP (maybe in combination with the used lens?) might not be the best benchmark here.
Point 4: high screen refresh rates are so high that your eye does not notice it at all. So, big difference with old/cheaper and new/more expensive.