I took a crack at it:

No matter what mixed strategy Snorlax adopts, there exists a scenario where Chansey can use its extra information to benefit from it. From the payout tables provided, the equilibrium against Counter Chansey is to use Body Slam p = 3/13 of the time, whereas against non-Counter Chansey it's p = 4/11. Deviating from either of these permits a strictly dominant strategy for Chansey in some scenarios, and because 3/13 is not equal to 4/11, that means there's no mixed strategy from Snorlax that forces a mixed strategy from Chansey.

From there, we adopt a strategy that maximizes what value we can get, assuming Chansey can use its perfect knowledge to change its strategy to best address our own. We end up considering a piecewise function:

1) where Chansey is compelled to always predict EQ when p < 3/13

2) where Chansey ideally adopts a mixed strategy to address Snorlax IF it has Counter; otherwise it predicts EQ; when p = 3/13

3) where Chansey is compelled to always predict Slam IF it has Counter; otherwise it predicts EQ; when 3/13 < p < 4/11

4) where Chansey is compelled to always predict Slam IF it has Counter; otherwise it adopts a mixed strategy; when p = 4/11

5) where Chansey is compelled to always predict Slam when p > 4/11

In case 1, the expected return is 5p.

In case 2, the expected return is 15/13, or the maximum value of 5p from the previous case. We can now eliminate case 1 as a desirable strategy, as this is strictly better than anything case 1 can give us.

In case 3, the expected return is 6 - 21p in the case of Counter, and 5p in the case without. 6 - 21p is always less than 15/13 in this case, which in turn is less than 5p, which in turn would compel Chansey to always choose the "with Counter" subgame. We therefore eliminate this possibility to maintain our expected return at 15/13.

In case 4, the expected return is -18/11 in the case of Counter, and 20/11 (the maximum value of 5p from the previous case) without. Chansey will always choose to play the "with Counter" subgame, which gives us negative expected return, so we avoid this as well.

In case 5, the expected return is 6 - 21p in the case of Counter, and 4 - 6p without. Both cases yield worse results than the previous case, which we already discounted, so we eliminate this as well.

So case 2, where p = 3/13, is our best mixed strategy for Snorlax. In this case, Chansey adopts the Nash equilibrium mixed strategy if it has Counter (which is to use Counter 5/26 of the time), just in case Snorlax gets any funny ideas. Chansey uses the strictly dominant "always predict Earthquake" strategy if it doesn't have Counter.

Note that the rate of Counter Chansey usage was not required knowledge here. If you want Counter Chansey usage to be fixed, the reasoning about cases 3 and 4 in particular would need to be revisited (but, importantly, this only affects Snorlax's strategy, as Chansey's is conditioned upon whether or not it has Counter). Regardless, I think the key insight is that Snorlax cannot force Chansey to adopt a mixed strategy.