You don't have to put .011"s on - especially if you don't plan to play it much. 0.010"s should do. And it's your custom guitar, not a Rickenbacker, so you can do with it what you want. Basswood is a bit soft, but if it's solid wood, it would have enough strength to support a Bigsby trem as the metal frame spreads the loading.
Personally I'm not a fan of Bigsbys, so I'd say leave it off, but have a look at some pictures of Ricks with Bigbys fitted and make your own mind up about the looks.