Rain isn't so much of a problem with hairpieces as it is with Toppik. Rain is the arch nemesis of Toppik! (Also, for blending issues with a hairpiece, DermMatch stands up to rain and showers, as long as you don't use shampoo.)

For that kind of hairstyle, I think that you are going to have to use glue, rather than tape, and glue is not as good for swimming as tape. Or so I hear. I haven't yet tried to swim with a hairpiece.

You may want to query others as to what glues are known to work for swimming, and you'll also probably want to experiment by soaking your head in the bath for an hour with the glue of your choice, to make sure that it actually stands up to such long durations of submersion.