Yep definitely no go for the tape!
Go to your local bike shop and ask for the old inner tubes they have from fixing punctures. They all generally have a big bag of them going for free to anyone who asks. Just cut the valve section out and use like that. Pull till you feel them 'bottoming out'.
TB111 will work perfectly - no need to try and degrease any wood with it. I've done a few glue-ups with it with lignum vitae no problems at all. lignum is the oiliest wood in the world by quite a big margin.
The main thing with it is perfect mating surfaces - no gaps.