Acoustic instruments are not really designed to have distortion, even if it has an electric pickup. Distortion is used mostly on completely electric instruments with no hollow sound body.
My Kala with Shadow pickups can't handle much distortion without sounding awful, since the pickup is designed to sound best with a clean tone and designed specifically to
avoid distortion. I imagine that this is the case for a lot of piezo pickups on ukes. Some pickups seem to handle distortion better than others - so perhaps that's something to take into consideration in your custom uke. Risa's nylon-string electric ukes use a passive Shadow nanoflex pickup, which seems to handle distortion quite well (at least on the Risa ukes.. not sure how they would work on an acoustic). Another idea is to buy an external pre-amp designed for electric guitars, and put that between the uke and the effects pedal. Theoretically, the pre-amp would process the signal so its more like an electric guitar's.
My Risa Soprano uke-solid and Bugsgear Eleuke worked fine with distortion, since they have no sound body to cause feedback. However, distortion would be even better on an electric uke with steel strings and magnetic pickups. If you want a uke that can handle distortion well, I advise getting a solid-body uke like from Risa or Eleuke. They don't cost
that much, but they will kick any acoustic-electric uke's butt when it comes to effects and distortion.
For your situation, you may still be able to get some distortion if you fiddle around with the settings. Try putting the volume levels on your uke's pickup (assuming its an active pickup with volume knob) to low, and setting it high on the amp.
I have also heard of there being sound hole covers on acoustic-electric guitars to reduce feedback when plugged in. I wonder if there are ones to fit ukes..
edit: Oh look there's a topic here about where to get one:
http://www.ukuleleunderground.com/forum/showthread.php?t=23675