Strictly speaking, it’s not an amendment to the tenancy, but an assignment of lease.
This is where one tenant replaces another with the landlords permission, but there is no change to the tenancy from a legal perspective for things like security of tenure and notice periods .
When I was a landlord, I had a letting agent who would charge for an assignment of lease. I, as landlord, always paid this.
I think your new landlord is chancing their arm in trying to get you to pay for this. You could try to refuse, but you might lose the property. It’s a tight market out there, so it depends a bit on how badly you want this.