FYI, Peter2772000,
The "titanium" on drill bits, is a coating, not the material of the drill bit itself.
Mostly it makes low grade drill bits shiny and stops them from rusting in the sea can trip from their homeland (usually China).
Titanium Nitride, Titanium CarbNitride, and a couple other "titanium's" get vacuum deposited onto the product to supposedly increase tool like, and reduce wear. Mostly, these days, it amounts to marketing gimmick.
If you ever get a chance to spend any serious time drilling holes, spring for some decent name brand drill bits from a tool supplier that deals with machine shop supplies (rather than the hobby guys, like Busy Bee, or most of the products sold at the big retailers like Ctire and Home Depot) Cleveland is good, there are lots of others.
Do you have a diamond grit knife sharpener?
You can shape carbide, slowly, with them. A dremel tool, and some diamond grit disks (steel disk with diamond grit on it) can do it too, also slow.
Either will allow you to sharpen up a small carbide tip of a masonry drill bit, if you can get one small enough to use for your hole.
Carbide end mills are sometimes cheaper than carbide drills in the small sizes, and can sometimes be fed down an existing hole to finish up what has been started in a case like this. Worth a price comparison, anyway.
Expect the drill or end mill to be trashed in the process. Contemplate the cost of a new part, vs. the cutting tools, too!
Cheers
Trev