Can the landlord insist on making variations to the lease whilst trying to extend a lease?

If you are using the Leasehold Reform, Housing and Urban Development Act 1993 then the landlord is permitted to make variations to the lease so long as they are in accordance with the provisions of Section 57 of the Act.

If you are extending a lease by private negotiation then there are no rules because you are acting outside of the provisions of the Leasehold Reform, Housing and Urban Development Act 1993. If you are not happy with this route then you can switch to using the the Leasehold Reform, Housing and Urban Development Act 1993