I voted for RZSS as there is a much greater selection of bird species of conservation concern kept ex-situ at Edinburgh zoo, a greater number of bird species kept and apparently more contribution to in-situ bird conservation...
I think this list is a little misleading for both zoos and really doesn't look accurate. ZTL lists 42 taxa in Edinburgh, five in Highland, whilst there are 13 in Wild Place and 46 in Bristol with seven taxa held in both collections, so that's 47 for RZSS against 52 in Bristol/Wildplace. Comparing the species (all info from ZTL) of conservation concern Bristol/Wildplace hold:
Extinct in wild: Socorro dove.
Critically endangered species: Blue crowned laughing thrush, Bali myna,
Edwards's pheasant (if we're accepting it as a valid species), Philippine cockatoo,
Endangered species: Eastern grey-crowned crane, Sumatran laughingthrush,
Visayan tarictic hornbill, African penguin, Java sparrow, Kea, Meller's duck, Sun parakeet, White-winged duck, Yellow cardinal
Vulnerable species : White naped pheasant pigeon, Victoria crowned pigeon, Palawan peacock-pheasant, Pink pigeon, Black-cheeked lovebird, Emei Shan liocichla, European Turtle dove, Mindanao bleeding-heart, Ochre-marked parakeet
Near threatened species: Nicobar pigeon, Fischer's turaco, Luzon bleeding-heart, Collared partridge, Crested partridge, European eider, Inca tern, Princess parrot
And Edinburgh hold:
Extinct in wild: Socorro dove.
Critically endangered species: Swift parrot, red fronted macaw, blue crowned laughing thrush, Bali myna, Baer’s pochard, Yellow-crested cockatoo
Endangered species: Waldrapp ibis, rockhopper penguin, capercaille (endangered within UK), Eastern gray-crowned crane, Madagascar teal, Western Egyptian vulture, White-winged duck
Vulnerable species : White naped pheasant pigeon, Victoria crowned pigeon, Emei Shan liocichla, Mindanao bleeding-heart, Snowy Owl, Southern ground hornbill
Near threatened species: Nicobar pigeon, chestnut backed thrush, Chilean flamingo, Satyr tragopan, Striated caracara
That's 25 species of conservation concern in Edinburgh vs 32 for Bristol, with a few taxa across both collections. Like you I've included the capercaillie which definitely isn't endangered across it's vast range but as Highland hold it with a conservation story I agree that it is very valid here (I think it would be thus fair to upgrade the turtle doves to CE if there was any good interp on this at Bristol, but from memory there isn't). Re-scoring the list giving 5 Arbitrary Conservation Points for a Extinct in the Wild species, and working down from there, gives RZSS 67 ACPs against Bristol's 77 ACPs. Bristol also conduct research/conservation activities on African penguins and Mindanao bleeding-heart doves. This may well be very inaccurate (good old ZTL...) and someone more knowledgeable about both collections could certainly correct me, but just on what I can find Bristol edges it on conservation and collection size. Numbers isn't the whole game of course, but I've voted 2:1 to Bristol due to them simply having species which interest me more (kea playing a big part in that) and holding some endangered taxa across both Bristol and Wildplace, allowing them to potentially play a bigger role in ex-situ breeding. Enclosures in both zoos are pretty good overall, with the penguin enclosures in both zoos being highlights, and I really do like Edinburgh's Baer's Pochard enclosure.