Thanks for your question
I am afraid that as this move of employment will not see you undertaking a relocation then any costs incurred cannot be exempted from tax and national insurance, or attract tax relief on them, as your travel costs and overnight stays are merely incurred in order for you to perform your duties, and are not of a temporary nature, on which a relocation will follow.
Whether your employer would be willing to cover this costs for a set time, during which you will then just be liable to tax and National Insurance on the payments covered (which is much cheaper then being out of pocket for the total cost) is something perhaps worth discussing with the employer.
Or whether they will help with the additional costs to live in this expensive area, which could then facilitate a possible move.
I am sorry the news is not more favourable.