payment plan

I am a self employed sole trader. I did a painting job for a building services company in the spring and I sent them a quotation prior to my starting. The work was completed and their client was delighted with my work. I invoiced the company for the  previously quoted amount of £9200 on 26th April. The operations director thanked me for my great work on site, acknowledged receipt of the invoice and promised to pass it on to accounts.

On 23rd June I spoke to the accounts dept. They said that payment had been authorised by a director but I wasn't then subsequently paid. I have made numerous calls but have since been unable to speak to anyone in accounts, put on hold, working from home and out of office excuses, promised calls that don't happen etc.

Then today after another attempt to speak to someone I received an email

"I can only apologize for the lack of response. I have been unwell recently so lots to catch up on. I won't lie this is the first time I have seen this invoice as far as I am aware. We obviously have every intention of paying it but would like to ask if we could set up a rough payment plan?
Would you be willing to accept £1000 each week and if we can do more some weeks we will?
Purely for cash flow reasons at the moment.
Many thanks"

This is now 84 days since my invoice and I'm not sure I trust them. I worry it will be occasional payments at their behest. As it is my four calls today have not reached anyone in accounts. I have a cash flow crisis of my own now and I really need payment in full.

How should I respond? Could their offer of a payment plan impact on me claiming expenses if I end up making a small claim?  ie. if I don't accept this offer am I rejecting reasonable efforts to resolve the issue.