Yes, you can pay your motor tax online nowadays. You should have gotten an email (or letter) about it.
PSV motor tax only has 1 option - 12 months. Having said that, when I got my new private EV, I paid 3 months private motor tax and when it was transformed into a PSV taxi, they refunded me the difference and then charged me 12 months PSV motor tax. So maybe you can pay 12 months PSV motor tax and get the remaining 6 months transferred over to your new taxi in 2024?
I think motor tax is paid into a central 'piggy bank' but it's administered by the local authorities. They probably just don't want to do the work, the lazy feckers.