This mantle was kiln dried for 9 weeks and has been sitting in our garage for at least 3 additional weeks so there shouldn't be a whole lot more shrinkage to it.
Some people seem to think that this is a fire waiting to happen but that's far from the truth. The folks that cut and dried this piece make their living doing this stuff and making mantles is a huge part of their business. Our previous mantle was also made of wood and heat/fire was a non-issue.
Thanks for the link 1minute!
Our paper birch sucks as lumber. It moves a tremendous amount of "movement in service" and has strong tendencies to warp, twist, and crack. Seasoning is very important, especially for indoor uses where stability is important. Allowing the slab to sit for several years will reduce a lot of those stresses, but not all. A couple months is as nothing to birch...
Kiln drying concerns me far more than an air-dried slab would. Kiln drying rushes the process to save money "with an acceptable level of degrade" and nowhere in there is there any hint of actually improving any mechanical property of the wood.
The tung oil will do nothing for wood stability as wood coated with oil will actually absorb and release water vapor faster than bare wood.
The last birch bark mantle I made was done by hollowing the slab and gluing it over a plywood box with "engineered" stiffeners inside the box. It will never move and the joints are virtually invisible.
If the log was cut last fall the bark should be good and tight. If it was cut this spring it likely will cause you problems down the road.
Superglue carefully run along the bark edges will soak in and help hold the bark in place, but it should have been done before any finish was put on. Use the thin stuff, not the heavier or thicker formulations.
You can just put it up and monitor it carefully and glue it back as it needs it. A cardboard piece to protect the bark and duct tape wrapped HARD around the edge to hold the bark in place should be able to clamp the bark in place until the glue dries. It will likely require some water to soften the bark enough to get it to wrap back into place.
You could also use a tiny piece of molding around all edges and use tiny brads to hold it in place. It will protect the bark edge also without looking bad. You should be able to make it yourself from the edge of a thin birch board.
Bottom line is I would suggest you just hang it and address any issues as they come up... and hope it was cut last fall...