Edit: Not sure what's with the formatting disaster. I wrote this in Word, so Bill Gates in the culprit, just not sure the mechanism that he used for the crime. 

I’m with you on the idea that
he won’t be x/r bluffing your flop cbet too much. I just think that the bigger
you bet, the less likely he is to x/r bluff, even though we’re talking about
miniscule, likely negligible differences in the affect on his x/r frequency
here. I think a bigger bet OTF, say 3k more, which is 1BB does have a meaningful
impact on how often he calls your flop cbet. And If we can invest 1BB more to
increase our fold equity, and secure the entire pot right now with no
resistance, it seems well worth it. Proposed hand that would be subject to
changing from calls to folds with this subtle sizing change are 22, 44-66, and
gutters like QTo, QTs, and 79s. I mean the fact is I’m splitting hairs here for
the most part, and our bet size here can’t be too crucial of a decision. Yet, I
felt compelled to speak those thoughts the subject! (*Rewatching the hand now,
and realized you did get check raised here! I hadn’t realized or forgotten than
in writing this post, so I should be bias free here, woot.)

Instinctively, this hand does
feel like it’s right on the cutoff of being able to want to check to be able to
induce, and call, some bluffs. I’m not too concerned with him bluffing with
better, as you said, it just seems like there won’t be many combos of such
hands. My biggest concern is him realizing his equity with 6 outs with KQ or
whatever.

Great points about the makeup
of his range. This is the key factor in our decision, as you said in wondering
if he has a lot of Ax here or not. KQ is a bit more likely to play different at
some point, but I feel pretty good about assigning him a near-full complement
of KQ combos. I’m not nearly as sure that he’s calling all KTo here vs your 3b,
even if he flats the initial raise. I do think his initial flatting range could
be a wider than it normally would, or wider even that we might expect, because
it could be his method of adjusting to BTN’s aggression; “BTN is raising so
many hand, I can call wider here in the SB now”.

Indeed we should consult the
machines on this one.

Okay, I think his initial
flatting range vs the BTN open is going to be on the wide side. I gave him
22-77, A80-ATo (AJo+ 3b/gets in), A5s-ATs, 87s-QJs, 97s-KJs (not certain here,
I’m not gonna assign him 100% of these combos, but I do think he’ll show up
with some of them given conditions) KTo-KQo, K9s, and QJo. Some notable
omissions/fringe hands: 67s, J8/Q8s, TJo, QTo, and some other Ax – suited or
otherwise. Of the hands I said he for sure has, that’s like 150-160 combos.

Taking the flop the flopped, and running your equity with Ad4c vs that range, we’re 35.3% vs 64.7%.   I don’t expect this to be an accurate or particularly meaningful number, as this assumes he’s called your 3b with all of these combos. It’s just for frame of reference. Let’s see what combos we think he’s peeling your 3b with.

Gone are all A8/A9o. Gone are
1/3 of his ATo combos (6 left). Gone are ½ of his KTo combos (6 left) Gone are
his A5s-A8s combos (6 combos of ATs/A9s remain). Gone is 80% of his small
pair  combos 22-77, that’s 30 total combos to start. I think 80% of them fold or jam pre, some will randomly peel. Of those remaining 6, I’ll give him one combo of a set (1). He has all 87s-KQs (20 combos). He has all KQo (12 combos). He has 2/3 of his QJo (6 combos). 2/3  of his KJo (6 combos). He has all QT/J9s (7 combos). He has half of his  T8s/97s/J8s for (call it 4 combos – all 97s for sake of calcs). He has all KT/AKs (7 combos). He has ½ of his K9/Q9s combos (4 combos)  75 combos remain. (ended up running the calcs with 102 combos since I couldn’t split combos in this program. I tried to estimate the weight. For example he has 3 combos of 333, but also an extra 10 or so combos or air broadway cards. Ad4c has 41.2% vs this range.

How is it best for us to realize that 41.2% equity, by betting or by checking? To help refine an answer to that question, let’s look at how his hands respond to a cbet. He has 25
combos that are air with no gutter, that are just folding. So that’s 1/3 right
there. He has tons of gutters that we beat. It’s questionable if he check
calls, check raises, or check shoves these hands (is it possible he check folds
some of them?) Q9ss (backdoor FD) has 40% vs us here for reference. Then he has
quite a bit of top pair and a bit of middle pair, 4 open enders, and a handful
of FDs.

Now I don’t know what this pile of unrefined information tells us, but I think we’re certain to get folds with 1/3 of his range here that is pure air, so that makes it clear that a cbet
is going to be profitable. I like cbetting to deny those hands a chance at
realizing their reasonable equity. If we can fold out some gutters, that’s great
too. And if he check calls his gutters at all, we can occasionally get to SD
and win. If he check calls JTs, we occasionally spike our 3 outs twice with an
ace. Things like these all improve the profitability of our bet.

A quick bit of devil’s advocate. I suppose we can realize a lot of equity by checking by capturing 2 bets from his 25% (6 out) hands on the flop. I do not have a good feel at all
for how that situation would play out, both in terms of villain’s approach/tenancies
in that spot, and what the math would say about all the various courses of
action that might occur. It seems a bit dubious though, and I would just assume
take the ambiguity out of it be cbetting. This also lowers our variance, which
is a tangible benefit.

Lord have mercy, cheers ifyou read all this mumbo jumbo. First round is definitely on me if I run into you at a casino or a poker table, haha.