I tried entering OP's mis-spelled reg, and attempting to pay.
There is a warning message "Be careful not to mix up the letters 'I' and 'O' with the numbers '1' and '0'."
1) OP says that "Twice it said my car is exempt"...
I didn't get any indication that the vehicle was exempt. I got this message:
We may not hold emissions details for this vehicle
Check your number plate is correct. We're not sure this vehicle meets emissions standards. This could be because:
It's a new vehicle - these tend to meet emissions standards. However, older vehicles manufactured before 2006 (petrol) and 2017 (diesel) generally do not meet emissions standards
There's a recent change of number plate
It's been registered as off the road (SORN)
We'll show you results for what you need to pay today, based on your vehicle being considered new. If your vehicle is not new you may get different results - without notice - when our records are updated. You should check what you need to pay each time you travel.
2) OP says that "The box you tick to pay the ULEZ charge was greyed out"
On the contrary, I had the same experience as OP's screenshots, which was the opposite. It is the exempt box [Free] which is greyed out. The box to pay charge can be ticked.
3) OP says "and I was unable to pay"
I was able to continue to put charge in basket, so seems like I would have been able to pay. If I so chose.
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It seems to me that if the OP knew the car was not exempt, and intended to pay, then they could have paid the charge using website.
No doubt would have been sent a PCN via the correct reg. number route. Payment would have been a defence.
So inability to pay via website is not a factor.
On other hand ff the OP did not know know whether the car was exempt. Then cannot assert that they were informed car was exempt as defence.
Only hope is to plead ignorance, and hope for mitigation.