Bath is a medium sized town, with two universities, in the prosperous south, hemmed into a steep valley and constrained by planning restrictions arising from its history. Housing supply is limited and typically expensive. The issues may have got marginally worse recently, but this has always been the context.
This means, to secure the housing you want, you need to be proactive, as others suggest: visit the agents in person to establish a connection and show credibility, sign up directly, nominate someone in the group to respond quickly to info coming in etc. Obviously this is not always possible because of group dynamics and is tough on first years who are often only just navigating independent living.
The other factor is flexibility - on location, property type and size (which also means the group composition may need to adjust and the candid discussions that go with that). The more flexible you are, the most choice and chance of success. There was a recent thread on Bristol accommodation that ultimately led down the same path.
I do have sympathy with the OP and their DC in relation to the late allocation of U of Bath first year accommodation last September; a possibly higher number of insured students coming through may have put strain on an already strained process. I think U of Bath will need to get better at managing expectations in that regard. For example, 1 in 5 first years are allocated rooms in the city centre (which does not necessarily diminish the experience but is a material fact), yet the promotion of the uni leans heavily on the campus aspect and lightly passes over the realities of accommodation.