You wouldn't use a masonry drill bit for engraving. Drill bits do superficially resemble engraving cutters, but they cut differently - straight down only. If you want a tool to cut sideways, it needs to be shaped differently. Diamond keeps its edge better than carbide. Here's an example of the sort of bit that would work (not that I recommend this particular site or seller): https://www.aliexpress.com/item/MUCH...8.4.149.irGzB5
You're right to be concerned about abrasive grit getting into the slides and screws of your router and damaging them. While dust collection systems or water blast can help with that to some degree, machines designed for cutting abrasive materials like stone or concrete are fully protected with bellows over the vulnerable parts.