I used to work in council tax many many years ago. I find it mad that you are offering money and they are not taking it. Completely crazy.
As we are at year end the 2020/2021 year pay what you can bearing in mind you will be starting 2021/2022 year from 1st April.
What should happen is that you agree and stick to (that is the important bit) any payment arrangement you make. They will likely want it in alongside this year's. So £70pm toward last year and whatever your payments are for this year.
Because you have been making payments I think there is a slight loophole in the system. What would happen usually (it probably hasn't changed) was you fail to make your monthly payment you get a reminder, you fail again at any point over that year you go straight to final reminder for the whole year to be paid, the person would usually ring, agree to monthly payments and pay.
If they didn't pay then they apply for a liability order from the magistrates court,(not a CCJ, nothing on your credit file) this is to prove that you are the person liable for council tax and it grants the council more powers to collect the money ie attachment of earnings or bailiffs.
Attachment of earnings is brutal and the rates are fixed. If you earn a monthly net salary of between £740 - £900 they take 7% and if you earn £900 - £1420 and they take 12%. It increases to 17% and 50% for figures higher than that.
You need to prioritise this any way you can, council tax is a local tax and they have far reaching powers to collect it in.
As a department we made every effort to help people when it got out of hand, we were kind and understanding. We had it day in, day out. You are not alone, this happens to lots of people. Just be prepared with payment plan offer.