You can invoice everything together of you can do it separately. If they are going to pay for the work actually done then you can deal with that separately if you wish but it really doesn't matter. Do have any specific concerns? When is payment expects for the work done? If that is likely to be soon then get that money in and sue for the balance. Happy to discuss further. I hope that helps. Please leave a positive rating so that I am credited for my time.
Thanks for your reply. It's not the invoice that matters but the basis on which the payment is tendered. If they make a payment for work done and say that the payment is in full and final settlement of all sums due then you cannot accept the payment as that would bar you from suing for the balance. So you should either sue for everything now OR wait until they pay the bit they aren't disputing and see whether they attach any conditions to that payment.
Yes, you can do it that way.
The email correspondence is sufficient for you to make a claim. The evidence of a lower rate for a longer contract is very relevant.