The one-shot case requires Bayesian thinking, not frequentist.

Cupholder managed to find an analogous problem in which the Bayesian subjective probabilities mapped to the same values as frequentist probabilities, so that the frequentist approach really gives the same answer. Yes, it would be nice to just accept subjective probabilities so you don't have to do that, but the answer Cupholder gave is correct.

The analysis you label "Bayesian", on the other hand, is incorrect. After you notice that you have survived the killing you should update your probability that coin showed tails to

p(tails|survival) = p(tails) * p(survival|tails) / p(survival) = .5 * .01 / .5 = .01

so you can then calculate

"P(red|after)" = p(heads|survival) * "p(red|heads)" + p(tails|survival) * "p(red|tails)" = .99 * 0 + .01 * 1
= .01

Or, as Academian suggested, you could have just updated to directly find

p(red|survival) = p(red) * p(survival|red) / p(survival)