I’ve had at least one report that with the polyurethane used by at least the first supplier, this is…not likely to work well.
The first customer who tried it said that removing the coating caused significant wood damage.
The original supplier was supposed to show me some stained pieces the last time I was in town…What they showed me instead had been spray-coated on TOP of the polyurethane. I was not pleased.
That said, you might have luck with a similar process, using a dye or paint. I’d recommend doing your tests on the inside of your enclosure where nobody will see it if things go badly wrong.