Welp, found a better route. There's a way to skip the double jump which saves a ton of time and cuts out the mini boss. Pretty sloppy. I'll try to clean it up.
And... I improved on the last run almost immediately. Still has plenty of room for improvement (that first jump is kind of embarrassing and the last room is ugly.) I think if this were cleaner it could be around a 3:15. If I could find a way to get through the spikes without the float then we could get really low.