One option is to cbet all of your range on flop with small sizing. That will keep your range uncapped. When we check flop we tend to say that we have mediocre sort of made hand. Something like KK, QQ, TT, KJ, QJs, JTs or weak Ax. Of course we can and should occasionally have hand like AA if we have check range here. AJ I would cbet on flush draw board but if it's rainbow it might be occasionally check also.
So if you want to check some of these mediocre hands be sure to balance that with some of the weaker Ax at least.
Issue of checking comes on turn and river. Let's say BB has A8. He can usually be fairly sure hand is going to beat most of your range. Though Q is not best turn card necessarily as it does improve QJ and QQ. But anyway point was that mediocre Ax can be fairly happy to bet turn big as it does beat lot of your made hands. I somewhat often go for big turn bet followed by big river bet in these spots. I don't expect much fe for turn bet but river barrel tend to get through quite a bit.
BB x-r size is something he could have just bet himself easily. So if player is good I think he can go for that line with good amount of Ax. In general if I check flop and bet turn I would go for at least half pot in your shoes.
On river all draws miss so big size make sense for BB. Your range have lot of mediocre made hands that are potential bluff catcher and hand like A8 beat most of those. Also by using big size opponent can have bit higher bluffing frequency.
So when we come to this river spot it stems from flop check decision. Is KK good enough to call? Is it enough for you to continue with Ax+? You do block nuts but you also block K♠: which is very bad. Hand like Q♣:T♣: would be better call imo. So I would say call at least K♣:K♦: but it does also matter a lot how your range looks like on river. As I said Q will improve hands like QQ or QJ that you would likely check flop also. Do you have hands like A8s or weaker in your range? If not then I think KK is just too high in your range and would probably have to be call. Of course call becomes better if you think villain is not doing this line with good amount of Ax.
But what happens most of time when we cbet small as range cbet is that BB call or fold. Raise comes rarely. Then we can usually decide if we want to check turn or barrel. Then if we do choose check we only have river remaining.
PS: In BB shoes I would have no donk betting range. It doesn't make sense as range advantage is clearly on early position open raiser. Same reason why BB should have very little flop x-r range because if he does raise hands like A2, 22 then his call range will weaken. Personally I want to have those mainly in call range if I play versus laggy player as that make my life lot easier versus multiple barrels.