Glass rod joining/welding: temperature instead of glueing?
I doubt this will work. For starters, from what I can see online, depending on what is in it, glass melts somewhere between 2700 and 4200 degrees F (1500-2300 °C). Butane burns at a temperature of 788°F (420°C). You also mentioned nichrome wire - but this too simply will not generate sufficient heat. Nichrome itself melts at 2552°F (1400°C), so it would not be able to transfer sufficient heat to glass without itself melting.

Even if you could heat glass up enough to melt it, thermal stresses can be significant. So you may find that the glass shatters if you did heat it sufficiently. Those craftspeople who do work with glass also have torches and furnaces that are capable of significantly more heat than you can get from any simply source that a handyman can provide.

Just use glue, perhaps a decent two part epoxy to join these pieces together.