Consider this spot:

Effective stacks 1000bb

BTN 3x, BB call
Flop Ah4c2c(6)
BB Check, BTN 3, BB call
Turn 7c(12)
BB Check, BTN 8, BB 32, BTN call
River 4s(76)
BB?

BB seems like he'll have quite a difficult time balancing a turn x/r'ing range here. Hands like Ac7, Ac4, Ac2, (Ac+2pr) seem like they clearly max their EV playing call tu call river. However, if BB never x/r turn with hands that can improve to a FH on board pairing rivers, on 12/46 rivers, he winds up in a situation where BTN's range will have a ton of FH combos and his range will have none. This results in IP going all in with NF+ and alpha bluffs and OOP losing the entire pot a significant portion of the time. It seems like our x/r range loses quite a bit of EV on these cards if it's not showing up with a FH here at least some non-0 percentage of the time.

Here we face our dilemma. On one hand, by never x/r'ing a hand like Ac+2pr we wind up losing a significant portion of a large pot on board pairing rivers. However, by x/r'ing those hands, we likely choose a lower EV line in a vacuum. My initial guess is that this is where mixed strategies come into play. Given we think it's very likely that x/ca is the higher EV option for Ac+2pr perhaps we choose to x/ca it 85% of the time, and x/r it 15% of the time. This way we're mostly taking the line that maxes that combo's EV in a vacuum while still allowing our turn x/r range to have some balance no board pairing rivers.

Is there a better way to look at these scenarios? I'm mostly wondering what how we're supposed to solve this problem in theory.