I would stay offsite for two reasons:
1. You won't even be able to enjoy that day at the resort. I don't see the value in paying Disney prices (or spending the points) when all you really want is a place to crash for the night.
We once arrived around 8-9pm, and stayed offsite. Our party was 3 adults and a 2yo child. Got a room at Sierra Suites with 2 beds plus sofabed for about $59, and it even included a continental breakfast. Everyone was aware of the plans, so we put our toiletries and a change of clothes in a separate bag and the rest of the luggage stayed in the car.
2. Most CMs that I've encountered go out of their way to find you a room that's to your liking when you check in. Whether you want something that's immediately available, particular part of the resort, particular view...they aim to please. But, if you check-in at midnight, you're probably going to get the last room of your class that's available. Could be a smoking room (or worse, a non-smoking room if you are a smoker.) Could be a handicapped room.
If you wait until the next morning, you are likely to be one of the first to check-in for the day and will have pick of the litter.
It would certainly be easier to plant your roots at Disney and not have to worry about moving. Guess it just depends on your personal preference.
Good luck.