Converting a contractual periodic to a new AST

Tenancy Types and Management

The property was on an NRLA AST tenancy Aug 21-Aug 22 for £1300 including an addendum that wrote in a 5% annual escalator. I failed to renew normally by issuing a new AST 2 months before it expired, and thus it became a contractual periodic tenancy at the same rent. I wish to now move it back onto a new 12-month AST at £1500, a rate comfortably visible as normal market price. Can I just service notice with a Section 21 and simultaneously issue a new AST for them to sign at the new price? Or do I need to instead issue a Form 4 from gov.uk/guidance/assured-tenancy-forms and if they refuse the new price doesn't that prevent me issuing a Section 21 as it could be claimed to be a "revenge" eviction? I am fairly confident the tenant will take the new price but I want to be sure of the legally most sound route before I action it.