The MP was unveiled as Reform's latest recruit by leader Nigel Farage at a rally, bringing the number of the party's sitting MPs to eight.
The new rules will be introduced in October and are describe as the most significant reform of driver licensing for 70 years.
The government has not confirmed what the cap will be, but campaigners have said they believe £250 a year is likely.
Emma Stone, Timothée Chalamet and Wunmi Mosaku are among the actors who could be nominated later.
Sir David and Lady Beckham were seen with their children Romeo, Cruz and Harper.