In theory, the fermentation process has already used up a lot of what makes the apple valuable as a nutrition source, leaving moisture (which they can get anywhere), cell walls/fiber (which is not as beneficial to chickens as it us to us - and something a free range chicken can readily obtain on its own), residual beneficial bacteria (acetobacillus primarily in making vinegar), and trace minerals/vitamins the acetoB didn't convert into acetic acid or dissolve into the now more acidic environment.
Its also a ready source for continued colonization by anything present in the environment which can either outcompete the acetoB (not much - part of why acetic acid fermentation has been used by human cvilizations to preserve food stuffs for at least 30 centuries), and the things that move in once the acetoB runs out of things to convert and can no longer support its colony size and growth. Aided by residual moisture.
If it were me, and I was trying not to be wasteful, i'd do one of two things:
A) lay the spent pulp out on sheet trays oir similar, bake dry, then serve it up as treats on the mulch/compost pile. If the chickens eat it, great. If not, its compost. A lot of studies show chickens dislike heavily acidic foods, so I would not expect it to be popular.
B) As an alternative to the bake dry and serve, you could also use it (since its chock full of beneficial bacteria) as the starter for fermented mash feeding. I'm not a huge fan of FF - the benefits don't outweigh the time labor investment for me - but its less time/labor then baking and storing. Since invention is primarily the province of the lazy, seeking a better way, this would be where I would start.
/edit forgot to add - Feedipedia.org on
apple pomace - be sure to read the nutritional aspects first. The tables show its not high value, but the aspects gives a broad overview of what to be on the look out for as potential problems and limitiations.