micromobility - Ebikes, scooters, longboards: Whatever floats your goat, this is micromobility
Ebikes, bicycles, scooters, skateboards, longboards, eboards, motorcycles, skates, unicycles: Whatever floats your goat, this is all things micromobility!
"Transportation using lightweight vehicles such as bicycles or scooters, especially electric ones that may be borrowed as part of a self-service rental program in which people rent vehicles for short-term use within a town or city.
micromobility is seen as a potential solution to moving people more efficiently around cities"
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That price will be very difficult to meet. For that money it'll probably have to be an AliExpress special. Get a front hub. You probably want a torque arm but I don't think you'll find any in that price range. You can get away without one if you run the motor at low power, like 250W. Unfortunately I don't think you can get a non-hazardous battery anywhere close to that price range. Perhaps look for LiFePO4 (LFP) battery since the chemistry is inherently safer and it's cheaper to make. It's heavier than standard Li-ion.
Otherwise I'd get:
With all that said, steep gradients can kill geared hubs so a direct-drive hub or a mid drive like BBS02/HD might be better suited. Most DD hubs are more powerful so you'd either have to use torque arm/a or run it at low wattage. In general aluminum frames can't handle a lot of torque at their dropouts without spreading or cracking. The rear dropouts can handle 500-700W when the nuts are correctly torqued. You can read about that on ebikes.ca.