If you've been battling I like either a weak bet or a shove on the river since he might be capable of shoving over top of a weak-blocker and he may be capable of calling you here down light if he thinks you're getting frustrated.

I agree with you that the river check is not great since this type of villain (I play very much the way you described villain) basically lives for playing the river in position with A high here...

By checking you're playing right into his game (he'll cry for joy when you let him show down Ax unimproved since he'd have been likely to call the river bet.

edit : I just read the other replies advocating c-betting. It seems to me that if you c-bet here, you basically have to be prepared to go to war when raised.

This is bad because :

-5x is part of villain's range
-he's unlikely to be 3-betting PP's higher than 88 preflop when he can just call and pwn you in position
-you can value-bet or check-call later streets very profitably after checking this flop (if he's behind he only has about 22% equity here so it's not like you're giving him a free flip)
-finally, you're likely to get raised by his air on a crappy flop like this which, for the reasons I just mentioned, puts you in a shitty shitty spot.