Check call or bet fold at this river?

£1/£2 weekday afternoon session.
7 handed.
Effective stack around £350.
Hero had 9♣T♣ at HJ.

LJ (young inexperienced player) limped, hero opened to £10.
Button (reg, semi-pro at least, tag, ) 3 bet to £35.
LJ called, hero called. If LJ had not called I would have folded as well.

Flop came A♦K♣9♥，hero hit button pair.
LJ checked, hero checked, button tank checked.

Turn was 5♣，now hero picked up a flush draw.
LJ checked, hero bet £40 into a pot of £108, button called, LJ folded.

River was T♠，hero didn't complete the flush but now got two pairs.
I thought for quite a while, till this stage it was still very difficult to narrow down button's hand strength. He definitely could have a lot of slow played AA, AK, KK, and even AT would be higher two pair than ours. I considered putting in a blocker bet, but struggled to determine the correct size to not look so obvious. After a minute or so, hero checked.

Button thought for 10 seconds then made a pot size bet of £180.

Hero again thought for a long time, called, hoping that villain had QQ or JJ and turned it into a bluff, but he had KK so we lost.

On reflection, I do not like my call at the river.
When counting villain's hand combinations, I could make the call given that I just need to be right once every three times, and villain was definitely capable of turning something like A3cc, 67cc or JJ into a bluff. However in reality at £1/£2 at this casino, you just don't see so many pot size bet over £150 as bluffs (maybe 10% of the time, and only from certain players).

Another question is whether it is better to bet fold or check call at this river?
If bet fold, what size?
If check call, what's the max villain bet I will then fold?