Thanks to Tod Shiner for suggestions and ESPECIALLY experience.

I'm coming at this as a complete novice (apart from owning a Mountain Bike back when ...) and at present am just tossing out ideas until a path forward becomes clear.

The greatest interest from tech-heads (me included) will be the Electronics Wizardry, and I'll post on this fairly soon.

Meantime, there are a few BORING but ESSENTIAL mechanical issues. HOW do you couple the cart to the bike? The traditional solution of a "tow-bar" with a coupling ball has long been accepted for cars etc, but I believe it's now out-dated, ESP for bikes. I've got some diagrams coming courtesy of The GIMP that should explain this and more ... just need some time between snacks ...

What about a hub or crank motor that freewheels with controller mounted on the mountain bike. Then make your own large 48v battery pack calculated for the extra load and distance you plan to travel. Put the battery pack on the trailer. Wire it with a coupler at the hitch. When you decouple you can ride the bike without all the additional weight of the batteries. 

I made a garden cart motorized with parts from a kids Razor belt drive electric dirt bike. Used 18650 batteries from old laptop batteries. Mounted everything under the cart. Used a 12v contactor and spst on the handle. Used it to walk from motel to beach. 

Thanks for the suggestions. Haven't seen 48V packs mentioned before. Is this a standard voltage for e.g. bikes and scooters? Yes, I'd planned to have everything in the cart except for a controller on the handlebar. Still looking at motor and battery options on AliExpress.

In fact you're right up there with the latest: "massless storage" and "structural batteries":

https://www.allaboutcircuits.com/news/Could-massless-storage-be-the-future-of-evs/

Good thought; though, as you say, for a DIY project a few off-the-shelf items like standard bike batteries is certainly the way to go initially. Looking ahead, however, hmmm ...

I bet the tubing that makes up the frame of that trailer could hold a heckuva lot of 18650s. Just saying.

Not really sure why I'm just saying that, actually. Can't think of a real advantage to stashing the batteries in a structural element like that, as opposed to simply building a battery pack and stashing it in an ammo can or something. But still,,,

Thanks for the comment. Yes, I'd wondered about that, hence the warning that it's "just an idea" that may not be practical. OTOH, if used as I envisaged just for cruising along once you're up to speed, and a power-cut switch wired to each brake lever, the safety angle may be solved. First up is numbers: max speed (20kph?) and power (plenty of motors about 100 - 200W on Aliexpress) and, as you suggest, cart drag etc. Then such things as drive-train: over-run clutches so that you're not turning the motors as you're pedalling.

It's a nice idea that can work very well. BUT: it might be very unsafe to push with too much power/acceleration, easily making the bike unstable. A nice feature might be to measure the extra drag from the cart, and use that as a factor on the pushing power.