What I personally would do is:
When heating a rod tip up never heat the blank. more then a couple seconds can damage the graphite. It melts pretty easy.
Do not use a cutter that will smash the rod, you’ll ruin the blank.
I use a razor blade and or a really fine hack saw blade and then file to cut my blanks.
You can reuse the tip you have if you can get the blank pieces off. You could try using a small drill bit and drill the blank pieces out.
Id find a rod tip glue not an epoxy. Should be no problem to repair this with about 10 minutes of work.
Any rod builder would be able to do this for you for cheap too.