I'm okay with checking behind here and calling a river bet regardless of what falls. But I guess this depends on how you feel about the generic player at your game. Would you expect a check-call on the flop with top pair on a monotone board or a check-raise? I think check-call is probably frequent enough that you can't reasonably rule out enough hands you're ahead of to make this a bet worth making.
I don't like the advice of betting the turn and calling a raise. The only way this makes sense is if you think check-raises here are incredibly rare. Because otherwise, you're kind of spending a lot of money for not a great return. You only make money when:
1) Villain has A
x/K
x.
2) The river is not a spade (which you'll pay off anyway).
Any other hand (other than total spaz hands) is ahead of yours when there's a turn check-raise. So you're going to put in 2 BB behind most of the time when you're check-raised. That's a pretty big deficit to make up unless you think it happens so rarely as to not have a major impact on the value you're trying to gain by betting the turn against A
x/K
x.