Stamps or permission letter?

When I apply for citizenship, can I include the one year of the INIS letter I got but only registered at the end of it's validity? The letter States that 4 stamp permission starts on the date of the letter but I didn't register it until 9 months later. Can I count those unregistered months as eligible time for citizenship and submit the letter as proof of it or the permission only started counting towards citizenship once I registered with the GNIB?

No. Stamps are your proof per instruction.
As a matter of fact, you will probably also need to explain why you did not register yourself with Garda in time. If there are 2-3 months delay, it is still a reasonable amount of time considering the unreliability of the online appointment system in Dublin, but only registering it at the end of your permission letter would sound fishy to me if I were the decision maker.