Ramadan Umrah is not a normal booking. Rooms near the Haram for the last ten nights are committed months ahead, and everything else follows the room.
Book the room first
Seats can be found late at a price. A bed within a short walk of the Haram in the final ten nights cannot. Fix the hotel, then the ticket, then the transfers.
What changes on the ground
Crowds change the day completely. Taraweeh, iftar and the sheer density around the Mataf mean short distances take a long time. Elderly travellers and young children need a closer hotel than they would in an ordinary month, not a cheaper one.
A sensible alternative
If the last ten nights are out of reach, the first half of Ramadan is far calmer and considerably less expensive for the same hotel. Rabi ul Awwal and the weeks after Eid are quieter again.