Never been a fan of the Eley in the field, no question they are quite accurate, but despite their reputation I've not found them any more accurate than many others. Target wise for sure!
Winchester and Magtech subs have a massive HP, just as accurate for me and expand well.
Remington subs are cheap as chips but awful, other than that they all tend to work reasonably well.
CCI ok but I use nothing but Subs in the .22 these days, and, for rabbits the odd crow or pigeon as mentioned in OP, no need for HV copper coated at all, subs do the job fine, and pass through less likely on al