I have 2 Great Pyrenees and years ago when they were pups and some we thought it was cute to have them in the pool too. I never had a hair problem from the mostly longer hairs and I'm sure that not all was caught by the skimmer basket or center drain or pump basket. It did cost me 2 liners for my pool though cuz my dogs have nails and vinyl versus dog nails, guess who won? You are brave..

By not fully priming do you mean when the pump is running? You may have a 'starvation' problem if no noticeable lines are plugged up.
On my 1.5 hp Matrix pump I will 'draw' half the pump basket empty if I close my center drain and only use the skimmer line and air will shoot out of the return. In my case this is likely caused by having all my plumbing 1.5".
Also be sure that you have no 'defective', broken, cracked lines. Air is easier to suck than water....I heard..

for a pump that is!
You did not mention if you get less pressure than 'normal' when this occurs.
If your pump is below water level does it fill up quick when basket lid is removed? If above water level does filling the basket with a hose help any?
To trace down something like this you need to be very thorough in checking the obvious things before taking anything apart, I think.
I hope my example of how "easy" it is to starve a pump helped you a bit.
Good luck!