Great question and this is something I have been working on lately.
I think your opening range might be a little loose. It is hard to believe. But if you cut out say 44- and three suited aces you go from over 19% to under 17%. I think ideally we should even be folding ATo here (although I admit I open this in the HJ often with mostly passive weak-tight players behind). You could replace ATo with a couple suited gapper combos (say 97s T8s). This will give you greater coverage on more boards. As is your range is 40% Ax which is a problem.
On this flop if you cbet all TP+ and all q-high and below you will be cbetting 38% which is not enough. (In general I would check TP with mediocre kicker but it's not in our range here.) The next hand I'd want to cbet would be underpairs -- if we get called we're probably not good and we are not likely to improve. (And bonus, if we do improve we are happy to play a large pot.) Betting 44- takes us all the way to 46.5% (hey what if we just didn't have these crappy hands to begin with?) Better but still not quite enough.
So let's bet our next crappiest hands which are weak Ax. A8- not hearts brings us to 53% which is about a reasonable cbet freq. But again these are hands we could have just folded pre as well.
So let's look at how our bet range fares if our opponent continues. Top pair+ makes up 42% of that so we're probably ok on that front.
But lo, our checking range is vulnerable! 117 combos of absolute crap! Nothing better than second pair, not one hand that can sustain a second barrel! We should have some top pairs in it. So it probably makes sense to always check your weakest top pair at the very least whatever it may be.
So we add KJ, and that's... 12 combos. Hoo boy we are in trouble. We're also calling with all 10 NFD combos. Because ace high is probably good against two barrels nearly as often as second pair, but we've at least got 12 outs to improve. But TP and NFD is only 22 out of 129 combos in our checking range. Still we've got no choice but to seatbelt up with a lot of second pairs. By the way most of our check-fold range is ace-rag coming back to bite us again.
I only want to call one bet with less than half our checking range. TT+, AJ+, AhT, Axhh, I guess 87cc/76cc too. 45.74%. Pitching nearly everything to a second barrel unimproved. Not fun
Thanks for posting this and getting me to do some work. The takeaway seems to be that having trouble finding a balanced cbet range is caused by being unbalanced preflop.