Point Of Sale Company Requesting Over $8,000 Won't Provide An Invoice

I own a restaurant and recently switched to a new POS provider. I intended to stay with my old provider until our contract ran out as not to pay any additional fees, but for the past 10 months our credit card readers have been near unusable. If someone tapped their card to pay it would shut down completely and reboot which takes about 5 minutes. If someone accidentally wiggled the card reader slightly when they inserted their chip or swipes to pay the card reader would shut down and reboot wasting 5 minutes of our customers time. The only remedy was signing a new contract for 4 more years and paying about $1,200 upfront for new card readers (on a subscription with a monthly fee as well) this is on top of signing a new contract for our already outdated POS equipment that I knew would not last 4 more years and lead us to another contract renewal to "fix" things again.

So I swapped POS companies and can't remember a time I've had a smoother experience even when I do experience a small issue. Our old POS company has a random account specialist emailing me math equations saying since I canceled early I owe $8,000 I know I will owe and even if it's that much I will pay and don't mind. What I do mind is that when I asked for an invoice for my records they just say "refer to the email for totals owed" and for some reason refuse to give me an invoice? I'd also like it to legotimized their numbers as one of the charges is my "estimated card fees minus Interchange fees multiplied by the amount of months left on your contract". They don't show any math on how they came up with that number it's just a number. When I use my credit processing statements and subtract the Interchange fees then multiply by the amount of months left I get a much smaller number than they do. Combined with not having an invoice to see how they have calculated their numbers I don't know how legitimate this is.

Also I'd like to have an invoice for bookkeeping purposes. Kinda sketchy to just tell my bookkeeper/CPA "oh yeah this $8,000 is for old POS fees but I don't have an invoice just trust me it's legit"

What should I do? Requesting the invoice just gets me "refer to my previous email"