Since you have the options I would just make a few from some different woods and see which you prefer, can’t hurt.
Here’s the description from the TMBR Replacement Axle page if that helps any:
[i]WALNUT is tough, slightly open grain wood. It is typically very smooth and long spinning, with very little tendency to burn out. Walnut is the stock axle material (for the most part).
OAK is very tough, with an open grain texture. It is long spinning, and the open grain gives a level of feedback on the string.
CHERRY is medium hardness, with a fine close grain. It spins smooth with a bit more response.
BIRCH is medium soft with a very close grain. It tends to be the most responsive species. It spins super smooth, but is subject to burn out over time. [/i]