Delaying a cbet

I've been trying to get away from an auto cbet regardless my hand, when I'm the preflop aggressor.  Last session, I tried experimenting with not "cbetting to take down the pot" when I have the best hand and have little to no chance of getting value on further streets without allowing my opponent to catch up a bit.  The problem, though, is that it seems that every time I try this experiment, it ends up burning me somehow...

I'm dealt AcAx in mid position and raise to $15 in my weekly visit to the Horseshoe.  I get a call from an older guy who has been very quiet up to the hand in question.

The flop comes Kx8c2c.  I feel like I have no hope, heads up, of getting any value by cbetting here.  I think there are very few hands he can have that can call a bet: Kings and club draws.  Since he's been so quiet, I think he'll often show up with a pocket pair -- 22 - TT, maybe JJ, in this spot.  Maybe he'll have a K: KJ, KQ, and maybe he'll have a club draw: KQ, KJ, QJ, QT, JT.  Since I hold the Ac, there aren't too many club draws he can have, though, given his tight image.  So, keeping to the title of the post, I check through.

Turn is Kc, and I haven't defined his hand yet.  However, he leads into me for $10.  Since I hold the Ac, and I'm a bit uncomfortable with the way the hand has played out (as this is an off-norm hand for me), I opt to call, ruling out folding and deciding against raising; I don't know where I am at this point.  Paired board, 3 clubs...

River is a blank and he now leads for $25.  I call and am shown QJcc.  My mistake is in calling here, because what am I beating except for a bluff catcher?  How often is my quiet opponent bluffing on a paired 3 club board?  $25 into a $50 pot - I need to be right 33% of the time in order to break even; I don't think I'm anywhere near right enough to pay that price, but yet I did it anyway.  Why?  I'm married to AA and feel like I'm paying a cheap price to showdown.  Correct; I am paying a "cheap" price to showdown, since I haven't fully represented my hand strength at any point.  If I'm really taking a critical view, I suppose that's why I make the river call - I've underepped my hand the whole way.  However, my opponent isn't really thinking about that; he's trying to get value from an opponent he views as trying to get to showdown cheaply.  In retrospect, I should prefer to check rainbow boards more often to allow my opponent to pick up a draw on the turn, and bet two-tone boards more often to "charge for the draw." 

Ironically, I think I lose less money on this hand played out the way above, vice the way I would "normally" play it: cbet the flop for $25, check through the turn and call a likely larger river bet since the pot is bigger by the river.  However, I probably fold the river so maybe it costs me marginally less?  I don't know; I'll continue to look for spots where I can underrep in theoretical way ahead / way behind situations.