I think your employer is on very shaky ground legally and this could be challenged but as you have less than two years service you aren't in a position to get heavy with them.
I would recommend that you join a union urgently. Most won't help with pre-existing issues but with an employer who behaves like this you are likely to require their help in future and it will be money well spent.
In the meantime, I would try to find out if any colleagues with two years service are in your position and whether they plan to challenge it. Encourage them to seek legal advice.
Assuming your employer already knows that you have only had one dose I would proceed as follows (I am not an employment lawyer though).
When you get to two weeks post vaccine ask your employer (by email or letter) if you can return to full duties as you are now fully covered until you are due for your next dose. They will probably say no unless they are getting cold feet but you have a record in case you need it and can show you were keen to get back to full duties.
Book your second dose ASAP, the waiting time has been reduced now.
After your second dose email and explain that you have now completed your primary vaccination course and make a request by email to return to full duties. If this is refused speak to your union for legal advice, but as you will still be under two years service your options are likely to be limited unless there is discrimination or something as a factor.
Book your first booster ASAP and again request that you are returned to full duties once vaccinated, and seek advice if refused.
Keep evidence of everything in case you need it in future. While it may not look as though you have any options legally at this stage this could change as things develop so think ahead. Communicate in writing, and keep copies. Keep in touch with other employees in similar situations, encourage them to keep records and to seek legal advice. If any have more than two years service then they may be able to mount a challenge that can help you all. Make it clear in all communication that you are willing to do regular LFTs as a stop gap.
I would also start applying for other jobs, this could be an attempt to push you out.
Also consider whether there are enough staff to do all the work or whether this will leave them short as this will affect your bargaining power. If it will leave them short an arrangement for you to pick up office based tasks for a colleague to enable them to pick up your client facing tasks with commission being split might be an option, especially if you make it clear that you are getting vaccinated and are keen to get back to full duties.