Linseed Oil & Wax compatible with Dye?

I'm about to start finishing a piece I've built, a small cabinet. The cabinet is made of a couple of different woods, and the two I'm wondering about are Oak and Ash.
In order to get more of a contrast between the two, I've applied a light dye to the Oak. I used Transtint colors (reddish and dark browns) in a solution of denatured alcohol. This got me the contrast I was looking for, and it looks great now.
I would like to use a linseed oil and wax finish. But my question is: Now that the piece is assembled, and the reddish oak is butted up against the un-deyed ash, will wiping on the linseed oil cause that dye to come back into solution and smear? That would ruin the piece.
I suppose I could always spray it, but I really didn't want to use a film forming finish on it. Can you spray linseed oil? Without blowing yourself up? (I'm just working in my garage). Any advice is greatly appreciated. Thanks!
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