Sorry, I cannot help with your die/tap search. My maximum diameter set is 1-1/2 inches.
However, in regard to your query on staining/finishing wood after soaking it in oil, here’s what I would try:
1- As a test item, use a piece of each wood stock species that you intend to use for your project. Then, using an oil-based stain only, apply stain to the entire wood test
2- Soak the pre-stained piece(s) in the mineral oil or raw linseed oil for your desired period (boiled linseed oil leaves a sticky residue when dry). When the time’s up, wipe the surface very dry with a rag. If the stain lightens after wiping, apply another coat to darken and re-wipe. When the colortone is adequate, wipe-on a coat of sanding sealer over the stained surface and let it dry (I always use de-waxed shellac, such as Zinsser Sealcoat). Finally, apply your finish topcoat, if using. You can also try the method, along with using an available tap/die set for the test. Doing so, will allow viewing of the full results on your intended project.
I’ve never tried this method on a tap/die project where soaking was used. But, after once reading a FWW article about finishes that will adhere to oily woods (Rosewood in my case), the instructions worked well for me. To overcome an adhesion problem, I wiped on 1 coat of Sealcoat on a pre-stained surface, followed by a clear coat finish. This eliminated any adhesion problems. The same article may also help you - at least with some of your finishing question.
In my opinion, staining before soaking, should not affect the pre-stain application because mineral oil is not a solvent - and I hope that I’m not proven wrong (after a lengthy soaking). Even when working portions of the wood with a tap or die after staining, there should still be enough oil left in the soaked wood (especially from the ends) to help with any needed lubrication. I'd at least give it a try.
Through my FWW site subscription, I downloaded the article mentioned above. The PDF is included with this message. Instructions highlight using oil-based finishes only, to assist with their adhesion. It may not fully help with your question, but the reading is another good learning tool.
Best of luck!