Wow this was a blast. If you polish things up you could have something really cool here. I can picture this being a perfect mechanic for a viral Roblox game. When you get into the groove it feels like Arsenal meets Doom 2016.
I think what would help the gameplay the most is minimizing any disorientation the player might have after teleporting. A wider FOV may help so that the player has an easier time finding the enemy that they planned on killing next. Moving the camera in a single frame might also be an issue. If it happens over a few frames with some motion blur and momentum, that could make the motion feel smooth and traceable to the eye. Finally, hitting enemies with hit-scan may be a better approach than a slow bullet,(at least of the main weapon) because then you'll be more certain if you got the shot so your eyes aren't surprised at the teleportation.