Well, you've already got her on the right track. Yes, I think skimmer socks will help and will probably be enough -- they shouldn't let hair through.

Burn-Out? Yikes! Bleach would, of course, by much less expensive. Even better would be to use some Dichlor initially to build up some CYA to at least 20 ppm and then switch to bleach and maintain 4 ppm FC in the pool (or 6 ppm if 30 ppm CYA is used). By not having any CYA in the water, the chlorine is harsher on the dogs than it needs to be -- not a disaster, but if she wants to minimize oxidation of the dog's hair then that's a way to go, especially if the FC has to be higher to not "run out" of chlorine (dog's use up chlorine quickly due to all the organics in their fur -- they usually don't shower before getting in!).

My "FC being 20% of CYA" is higher than for residential pools as an extra safety margin for disinfection since dogs can introduce a lot of bacteria (saliva, etc.).