Here is a toyish game that you can use to deduce the answer. Lowball single draw. One player has a face up weak hand. The other player has a face up good one card draw that is a small underdog to outdraw the pat hand. A pot sized bet, but no raises, is allowed before the draw and after the draw. Only the one card draw is allowed to bet. Or he can check.

That one card draw should bet the first round bet if the chances he will make his hand is above one third. (Because after the face down draw if he bets with a frequency a hair below one an a half times his chances to hit, the other guy should fold). But what if he is dealt two face down draw cards and can only check once or bet the pot once? When called many pots will be split. Does that change his correct before the draw strategy? If so, how?

I've never tried constructing hypothetical nash betting ranges when chop pots are involved, so this may be incorrect. But from what I can tell, you'd start by betting all your scoops (obviously), then add twice as many of your chop combos. This provides your opponent with the indifference ratio of 2:1 value to bluffs. (If you don't have twice as many chop combos, then you'd include 1 pure bluff combo for every 2 "leftover" scoop combos)

In the situation where you have at least twice as many chops as scoops (which holds true for any equity < 50%), your additional captured share of the pot on the river (by "bluffing" with chop pots) ends up being equal to your % of scoop combos.

Additional share by bluffing = Scoop combos = (raw equity)^2

Total share of pot = Additional share + raw equity = e^2 + e

0.5 = e + e^2

e = 0.5 * sqrt(3) - 0.5 = ~0.3660254


So by changing post-draw rules, his pre-draw strategy changes to checking any draws with equity between ~0.333 and ~0.366 as they no longer capture >50% of the pot on the final betting round as they did before.