I'm not overly surprised to hear that a person can handle stairs but struggle with stepping into a bath, they really are two very different actions, also the availability/positioning of handrails etc is likely to be very different.
I'm aware that you have posted on my 'Ask an Occupational Therapist' wall and I was try to respond more fully as soon as possible.
Jean Merrilees BSc MRCOT
You can read more of my posts at: https://community.scope.org.uk/categories/ask-an-occupational-therapist
I have replied on my "Ask an Occupational Therapist" wall but for completeness and also for reference to others that might be interested in the answer I am copying my response below:
So, the first thing I need to stress is that it is impossible
for me to pinpoint exactly what adaptations or equipment would work best for
you. Making that judgement is going to require an in-person assessment by an
Occupational Therapist (OT) who visits you at home to explore with you what the
difficulties are and discuss what changes may be possible. You can request a
visit from the Community OT by contacting the adult social care team at your
local authority (council) or your GP can contact them on your behalf.
Presumably the equipment that you are currently using was
supplied by the OT but if it isn’t working for you then they need to come out
and re-assess and have another think.
In many areas getting Community OT support is challenging
due to long waiting-lists and insufficient departmental budgets to meet local
need, but do persist!
People who have the financial means to pay for their own
equipment/adaptations sometimes decide not to wait on the Community OT and
instead engage the services of on OT in private practice: https://www.rcotss-ip.org.uk/find
Either way, it is a good idea to have an idea of what
products are on the market prior to the OT visit. AskSARA is an interactive
online information resource about disability equipment that allows people to
work through questions to find equipment targeted to their needs: https://asksara.dlf.org.uk/?auth=sara5
My suspicion is that if your vestibular disorder is one that
is going to persist that ideally you may need to have a walk-in shower with a
seat, rather than a bath. Potentially this is an adaptation that can be funded
via a Disabled Facilities Grant (DFG) https://www.scope.org.uk/support/disabled-people/disabled-facilities-grant
The Community OT will be able to advise if you meet the criteria for a walk in
shower both in terms of assessed need and the financial means test.