Might not be convenient for everyone, but one alternative is to book a night at the Sofitel together with parking.
£230 for one night and 4 days of parking.
Only about £80 more expensive that what short-term was quoting me and I get a few extra hours of sleep before an early flight.
The Sofitel is closed until further notice, They have cancelled all of our July and early August bookings. The T4 crowne plaza which has parking is also not available to book in July or August.