Paint Bleeding into Wood Fibers

I'm engraving into wood and I'm having trouble with paint bleeding into the wood fibers. Here's what I did:

1. I put several coats of polyurethane finish on the wood. I'm using maple. The surface is sealed.
2. I engraved my image and lettering. The image is 0.01" deep cut with a 60 degree vee bit. The lettering is 0.05" deep with the same bit.
3. I filled in the image and lettering with flat black latex paint. I covered most the surface and wiped most of the paint off.
3. After the paint dried, I sanded the remaining paint off the un-engraved surface.

It looks pretty good from a distance. When I look closely at the image and lettering, the paint has bled into the wood beyond the edges of the engraving. I assume it's capillary action into the wood fibers.

Does anyone have any suggestions how to stop this bleeding?

I'm a woodworker so using wood is natural for me. If the solution is to use something other than wood, what do others use?

Thanks,
Dick