NO OIL!!!! That would prevent anything from sticking the two of them together.
If you have ever dried down a batch of rehydrated hide glue, you would see that it reduces in volume even worse than PVA glues like the TiteBond line. If you had perfect wood to wood fit, hide glue might be find, but for any gap filling, forget it!
Post pics, win or lose, we can all learn from what you got going on!