You should definitely play with the elmoto spreadsheet and see what you can do. Here's an example set up that should be within your budget:
ME0709 motor / Alltrax 7234 kit w/ contactor: $900
Magura Throttle: $50
GBS 40Ah, 72V lithium battery pack: $2300 (includes BMS)
Sprockets: $15-100 (depending on if you need a new rear)
Misc cable and such: $100
Materials for battery shelf, motor mount, other odds n ends: $200 (estimate)
Total: $3700 (USD)
This would easily get you a top speed of 70-90 kph, 40 km range. Way better than scooter performance.
You could save about $1000 with lead acid, but keep in mind they weigh twice as much. Lithium will last at least 4 times longer, probably more.