If those are the only two options, then you've gone mad :-)

L(H|e) is defined to be P(e|H) (which, yes, was a confusing and bad plan).

Reporting "the probability of H given the actual data e" would not work, because that requires mixing a subjective prior into the objective likelihoods. That is, *everyone* can agree "this sequence of coin tosses supports 'biased 55% towards heads' over 'fair' by a factor of 20 to 1", but we may still disagree about the probability that the coin is biased given the evidence. (For example, you may have started out thinking it was 100 : 1 likely to be fair, while I started out thinking it was 20 : 1 likely to be biased. Now our posteriors are very different.)

The reason humanity currently uses p-values instead of Bayesian statistics is because scientists don't want to bring subjective probabilities into the mix; the idea is that we can solve that problem by reporting P(e | H) instead of P(H | e). The objective measure of P(e | H) is written L(H | e).