Isn't the controller integral to the rack battery? How do you know it's the controller that needs replacement? I suspect it won't be available as a part on it's own.
Pulse was the budget end brand of Ebco in Warwickshire - e-bikes imported from the Far East under their own brand name(s). At that time they used TranzX motors, controllers, batteries, displays. Ebco don't deal with TranzX anymore. You could ask if they hold any old stock, but I suspect support is now zilch. There are no other mainstream importers/suppliers of e-bikes using TranzX drive systems now in the UK apart from Raleigh (and only in a limited way)...
I recently spoke to a dealer repairing those Raleigh e-bikes that use TranzX parts - he advised a 6 month delay obtaining many TranzX parts due to Covid/Brexit.
Holland Bike web-shop (and other EU suppliers) still supply TranzX parts. They may be able to help, but I suspect you may need to buy the whole battery/controller arrangement.
The other option would be to strip it all out and replace with current day generic parts - others here could likely advise on that possibility.