Check into wood that already has documentation. If you have the paperwork on wood that says it meets the various qualifications and that it’s not illegal then you can ship a lot easier.
In the musical instrument world Cole Clark and Maton guitars use native Australian woods and are shipped to the US all the time. I use to be a dealer for Cole Clark.
Some times there are loop holes for “Finished products” also, if the wood is known to be a safe wood (not harbor insects or disease), and not endangered, then it may be fine to ship these to the US.
Just a couple other options that just came to mind.