Hi Brian, what is the stated maximum continuous current output for your new battery?
Many cheaper eBike batteries are only really suitable for use with 250~500W motors
(approximately 5~10Amps at 48V).
If your battery cannot supply high current
(~30Amps) it might keep cutting out when your MPIII is under maximum load.
Please Note, it would be extremely easy
(and also very dangerous) to inadvertently connect the battery directly to the 240V mains using a standard computer style IEC C13 mains connector
(especially if the charger input uses the same plug) instead of the charger output lead.
I am pretty sure that it would be illegal to sell a battery with a IEC C13 mains connector being used for a low voltage DC connection in the UK.
If you decide to fit the new battery, keep an eye on the lead and connector on the new battery to make sure it isn't getting too warm under load. You may also find that the power is much lower than before even if the battery does not cut out under load.
If you know exactly what cells are used in your old battery, you could hopefully find out how much it would cost to get new ones to rebuild the old pack, which may or may not be cost effective to do.
If the cells are spot welded inside the pack, it would require a spot welder to rebuild the pack with new cells, as the cells can sometimes be damaged by the high temperature required to solder the connections.
Alan