Am i able to use the blazing bowstring ability on an attack with a bow alongside the rogue's sneak attack ?

and if so I would roll double dice on not only the weapon hit die, but the sneak attack die & the blazing bowstring die. 

I am a lvl 4 Rogue. My base damage is 1d8 + 6. 

If everything plays nice, a sneak attack crit + blazing bowstring & sharpshooter feat would make my max output equal to

6d6 + 2d8 + 16

Or am I missing something here ?

Thanks

AFAIK, that's the way it works.

I'm pretty sure in the show itself Vex lands sneak attack damage with it more than once. It's a fairly rare item, so getting it at level 4 is kind of surprising lol but other than your DM being generous there's nothing untoward going on.

Yeah I realized that it generally costs 2500-5000 gold, I got mine for 250g. Both the bowstring and other homebrew magic item I have I didn't really ask for, and other players are getting a lil upset over so I might talk to DM about changing them. 

You could always offer the bowstring as a gift to an NPC in exchange for a story-advancing reward. That removes the source of tension and provides a gift that the entire party can appreciate.