The Jones bars are standard diameter so most grips should work unless they are an odd size. The Ergon grips with bar ends might be a bit awkward with the 45 degree sweep. Cable length is always a consideration when swapping bars. Control cables (brake & shift) aren't too difficult or expensive to replace if necessary. The ebike wiring harness is another story. Some brands offer extender cables which just plug into the existing connectors. It is sometimes necessary to piece out the wiring which requires a bit of skill. Doing so may also void the bike warranty. Your LBS may be able to help you here.
To get an idea of the fit, I removed the grip and controls from one end of my existing bar. I held them in the approximate position where they would be on the Jones bars to see if there was enough slack in the cables & wiring. The Jones website gives bar dimensions. In my case, the old bars were the swept back mustache style so there was plenty of slack. I don't own a BH but from pictures, it doesn't look like there is much cable slack on the bike to work with. Again, your dealer may be able to advise you.
I'm 5 foot 7 with short arms. Hence I would like the backsweep. Your son has this same bike - do you think the Jones would fit on it? It looks like the display might be limited in cable length. I agree that I don't want to mess with getting new display extension wires and voiding the nice 5 year warranty.
The problem looks like the control panel has a limited cable length to the main display.
With the Jones bars, the control unit might not sit close to the end of the bars where the grips are?
I own a BH Rebel 5.5 with the Yamaha PW-X system and there is plenty of slack in the control cables and wiring to add a riser stem and swept-back bar.
There is no need to worry about extender cables or rewiring any controls. Hope this helps...