Dyes are your best choice. Alcohol or water-based are only minimally affected by oil finishes, or, perhaps we should say "oil-based"😉. So any of the aniline types become a possibility. You don't mix them with the oil, of course, rather apply them to the piece and finish afterward.
I consider pigment stains, the kind most normally associated with flat woodworking, to be unsuited to turnings. They're dust suspended in oil, and give an entirely weird-looking result. If you're looking to hide the wood, tempera color is a good option.
I'd lacquer or shellac over dyes. The amber color of most oils will change the non earth tone shades unpredictably.