[rofish]

Diemai,
I simply do not understand why you have to go through such a complicated way of gluing the wire to the wooden lip.
I see the reason why you have to put first a little glue on the wire, to keep it in place, and then continuing the gluing process. But to speed up things, I would simply use some superglue to make the wire stay in place. Superglue dries instantly, so it saves time for you.
Then, the problem with the tape. On one hand, you say that bubbles in the glue must be avoided. On the other hand, you say that if you press the glue from above, using a toothpick, it is most likely that the slot is not entirely filled up with glue. I think this is quite a logical result, since the air is trapped between the tape and wood, and since you press the glue from above, with a toothpick, and through the slots in the wood, I assume, the air has little chance to escape through the unique way out - the small slots, made for the 2 peaks of the "M" shaped wire to go through the wood.
I think there is a simpler way to glue the wire. After the wire is glued with a little glue in place (either superglue or your 2 component glue), you find a method to block out the 2 slots, so the glue would not be able to go through because of gravity (you put there small pieces of wood, or use a little of your 2 component glue the moment it starts to harden).
Then, you glue the wire in 2 steps, once from above, and then from the other side. If you glue the wire this way, you can avoid air bubbles, which have a natural tendancy to go upwards, through the glue.
Another method would be that you use a very soft wire, which you can form directly on the lip, and in this case you do not make slots, but just holes, a little biger than the wire, and so the glue could not go through the holes.
I have to try your idea with your egg box. Very clever one.
Recently I have used a similar squeeze tube glue, similar to yours, to glue in lips. It is a 10 min. epoxy, 2 components, transparent not clear, as I presume yours is. Don't know if it would be suitable to glue the twisted wires in. After cure, unlike most epoxies, it is still soft.