The fact that two other recommended places to stay were full explains why the room rates for the Z hotel on weekends are high - simple demand.
J@p28al, can you rearrange your stay in Bath so you're not there on a weekend?
It will make it easier to find a hotel with a room available at a cheaper price.
Sometimes when we think we have our itinerary planned, we have to juggle things around. My husband and I do a lot of rearranging when making a trip plan for England.
Can you rearrange your schedule so that you hit Bath mid-week?
You don't give the specific date you're hoping to stay in Bath, but if it's this year, know that a lot of major events throughout 2025 are planned to celebrate the 250th birthday of Jane Austen.
https://visitbath.co.uk/inspire-me/literary-bath/jane-austen-bath
The busiest time will be during the Jane Austen Festival, Friday 12th to Sunday 21st September 2025. This will be the height of the fun and largest crowds.
https://janeausten.co.uk/pages/festival-home-page
If your itinerary puts you in Bath at these times and you are not wanting to be there specifically for these events, I would say, avoid Bath.
Can you put it in somewhere in your itinerary at a later date when hotels are showing vacancies and reasonable prices?