Is there a way to put the whitening gel in my clear retainer tray and avoid contact with my gums?

First, make absolutely sure that the bleaching tray does NOT touch the gums.  If they do, your dentist should easily be able to remove that excess to prevent the bleaching gel from getting onto your gums.

You may not be doing anything wrong.  Some people are very sensitive to some of the whitening material and have reactions no matter how careful they are.  You might try one more time and use the absolute minimal amount of gel.  If this still irritates your gums you probably won't be able to use that gel.  Good luck.