new2it I would still ask despite being desperate for a job - they obviously want you for this job and requesting a review of the starting salary isn't going to change that in itself.
I don't know about LEA salary structures, you are starting at the lowest point in the band your job has been allocated, and it may well be that that is automatically the case when you are starting from scratch, but as they have told you that you can request an uplift, it is obviously possible to get one.
I would have a good look at the job description and person spec for the new job. The best way to justify an uplift in your starting salary is to point out what additional experience/expertise you are bringing to the role over and above the requirements of the job. Obviously I don't know how different this new job is to your old one, but I'm presuming that most of your skills and experience at the previous (higher-paid) job is transferable and relevant and therefore a slight uplift is not too outrageous a request.
I'd say something like
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Thank you very much for the offer of the post of [job], I am really looking forward to starting work with the team. This role does represent a drop in salary for me based on the starting salary I have been quoted. I would like to request that consideration is given to adjusting this to reflect the skills and experience I am bringing to this role.
[then talk about all the marvellous skills and experience you are offering]
I will be happy to discuss this further or answer any questions you may have with regard to this request, and I am very much looking forward to working with you.
kind regards
new2it
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That way you are asking nicely, you are reminding them about all the wonderful things you are bringing to the role, and you are not burning any bridges.
You could phrase it more harshly, calling their bluff slightly and making out you will not be interested in accepting the role unless they match your salary, but I really don't think you are in a position to do that, I'm sure you agree. But just because you are not in that strong a position doesn't mean you can't ask nicely.