When feeling up to it.
How did you connect that controller to the charger to get that part to work?

Is that a Bosch PowerTube 750 battery?
Does pressing the ON-OFF button on the battery make any difference?
Other suggested wiring configuration:
View attachment 363010

The power supply (36v) wires went straight to the 'IN' + & - terminals and then via the 'BAT' terminals direct to the battery.
Yes, Bosch powertube 750.
Yes, pressing the on-off button switches on the battery for about 3 seconds when disconnected from a power source. Voltmeter shows battery voltage for those 3 seconds.
I'm just about to try connecting up your new circuit. Will report back.

 

New circuit connected, checked and tested. No joy. I think I can see why. The power supply needs to see the positive from the battery before it will switch on. This circuit has that connection isolated initially by the open relay. So no power can come in so no relay operation.

 

The power supply (36v) wires went straight to the 'IN' + & - terminals and then via the 'BAT' terminals direct to the battery.

Like this and this worked?

Try this arrangement:

 

Well, that works... sort of. Applying power to the power supply and pressing the button causes the dpdt relay to close, board relay to close and display to light. But no relay in the charger. But then after a while it clicks on and the system works as desired i.e. it charges to the cut off voltage, cuts off and does not restart. However the battery then shows a single led error:
'A single, blinking LED on a Bosch PowerTube 750 (often the middle light) typically signals a charging fault, often related to temperature, connection, or a BMS error.'
I'm back on duty with dying relative tomorrow so this goes back on the shelf. I think we have made progress and I really appreciate and admire your stamina in trying to get this working. We're pretty close to what I was hoping for.

 

I'm back on duty with dying relative tomorrow

Sorry to hear that, God bless.

 

There is a reset procedure for the battery using the power button, might help.
To reset a Bosch PowerTube 750 battery, remove it from the bike frame and press and hold the power button (located on the battery casing) for 10 to 15 seconds. The LED lights will flash to indicate the BMS has reset.