This answer may be helpful, and/or it may be annoying. Your welcome and my apologies at the same time :)
One thing to remember when using a normal distribution, is that it has a set of sufficient statistics, namely the mean and variance. What this indicates is that only the mean and variance matter in the inference. Any property of your sample besides the mean and variance will be thrown away when you use a normal distribution.
The statement that the "population is not normally distributed" is a bit of a misnomer - the population is not "distributed" at all - there is one and only one population (imaginary data sets and alternate worlds aside). It sounds like what you are actually saying is that your knowledge of the population consists of something other than the average and variance
So presumably, the only thing to do is to state what this extra/different knowledge is. Perhaps you know the skewness (or you know the skewness is important/relevant for the analysis, and not "noise").
I would suggest that you simply calculate the probability that your hypothesis is true, conditional on the information you have. This would include the data, and whatever "structure" you claim to know about the population that makes it non-normal (something other than the mean and variance of the population). So call your one sided test $T$, then you simply calculate:
$P(T|I)$ is the prior probability for the test being "true" or "successful" (what did know about the test prior to seeing the data?). $P(D|T,I)$ is the "model" or "likelihood" and is similar to a p-value (How likely is the data you observed, given the test is true?). And $P(D|I)$ is often called the "evidence" (how well do any of the hypothesis predict the observed data?) - this quantity does not need to be explicitly assigned, as it can be derived from the requirement that the probability must add to 1.
The good thing about this method, is that probability theory will "construct the optimal test for you". You just need describe your prior information, and then simply do the mathematics. Now you may find that a bootstrap may be necessary in order to evaluate some mathematical formula - you may find that you should do the wilcoxon test - or probability theory will construct a test which is better than either of them (in terms of that type 1 and type 2 error you speak of).