Just got back from the wood shop. Guy there suggested dying epoxy, filling the engraving, then sanding the epoxy flat before applying oil.
Just realizing now that that will eliminate the 3D nature of the engraving, but given my time crunch, it might be the best plan.
Edit: just did a trial run with some random acrylic paint and BLO. Really thought the oil would affect the paint, but apparently that’s not the case.
This scrap was roughly sanded with 120 grit for just a few minutes. I expect it’ll look great with proper preparation.