I'm not suggesting hub motors.
When we convert a pedicab, we attach a second sprocket to the differential housing, and drive it with a chain from a shafted motor. You say it can't be done, but I've done it more times than I can remember. Between the conversions in the company fleet, the ones we've sold to other operators, and the ones others have reverse engineered from our design, there are hundreds of such kits in service right now.
Patent Pending, Stainless steel construction. This is a complete bolt on (zero welding needed) system to attach a cyclone motor to pedicabs that use a 2 bolt axle bearing plates and have the rear brake mounted behind the rear axle. (all Classics and most San Diego pedicabs) This system improves...
precisionpedicab.com
I don't think our specific kit would retrofit to a Hase trike, but I have no doubt at all that a similar setup could be made to work with your trike. Without major modifications to it.