Motability are very rigid about not letting anyone order earlier than 3 months before the lease end. The added problem you may have is that many dealerships don’t open on Sundays! If they do, the order process is automated and takes minutes, as all the system is checking is that you are eligible to order, so if you place the order on a Sunday it should be approved that day.
If you do order on the last day of the month, ensure that the dealer agrees that if the AP drops the next day they will cancel and re-order on the Motability system. It only takes a few minutes and doesn’t affect what they receive.
If you order before 1st July you will hopefully miss all the extra charges. Motability honour the prices based on the day you order, so you don’t need to worry that you will be hit with extra charges mid-lease. You won’t, but you’ll get the full hit with the lease afterwards.