I need to leave the Hilton Niseko and be at Chitose airport for an internal flight to Osaka during a holiday (Feb, date of flight 21st February) but the flight is a morning 1040 am flight.

I think this means I need to be there at the airport by 9.30 am.

Theoretically, you can check in until 15 minutes before departure.

By train from Kutchan, it seems you can be at the airport early enough:
http://www.jorudan.co.jp/english/norikae/

You could also rent a car, but give yourself a buffer for eventual delays and the procedure of returning a car, which takes a while because the rental outlets are quite far from the airport:
http://www.japan-guide.com/e/e2024.html

I was told I needed to be at the check-in at least 1 hour in advance. Is this not the case? I have luggage to bring along.

I was told there is a courier service to transport luggage? But I'm under the impression they only operate from airports.

Is there such a service to deliver my luggage from Hilton Niseko to my hotel in Kyoto (Hyatt Regency) overnight?

I am staying in a ryokan that evening when we arrive in Kyoto so there is no need for all the luggage and it would be easier to navigate train to plane to train from Osaka to Kyoto without the bulky luggage.

Check-in is possible until 15 minutes before departure. Boarding is possible until 10 minutes before departure. If you have luggage to check in, I recommend to add an additional 5 minutes.

So, no, you do not need to be at the airport one hour before departure. But especially if you are not familiar with the airport, it is recommended to be there about one hour before departure.

I was told there is a courier service to transport luggage?

Yes, details are here:
http://www.japan-guide.com/e/e2278.html

But I'm under the impression they only operate from airports.

No, they operate from anywhere to anywhere.

Is there such a service to deliver my luggage from Hilton Niseko to my hotel in Kyoto (Hyatt Regency) overnight?