I am not sure you understand the nitrogen cycle, I may be wrong....but, you may have cycling confused with aging....
IME as long as you have some type of oxygen source along with food for the nitrifying bacteria you can cycle a 1g container, so a 2.5g is possible IMO/E
For your trip, how long are you going to be on the road?
To keep your cycle you will need to get some type of air pump that will hook up to the cig lighter or battery operated air pump, and once you have the tank in the car..... cover the substrate with 2-3 inches of water and float the filter media so that the air stone is circulating around it, if you are going to be on the road more than 12 hours.
Also, fill some jugs with your source water and take with you so that when you get to your new location it will be safer/easier to use the source water in the beginning and then with your water changes you can get him used to the new source water, sometimes the pH can be different and that can cause some problems with the fish and if you take some water with you it will make things easier on both of you and if it is an over night trip you can use that water for a water change.
It is important to keep the Bettas water temp stable too, I like to use small coolers lined with something, packing peanuts, newspaper etc.....to avoid extremes
IMO....I think you need to make the trip safe for the fish (I know you will) and not worry about your cycle, in a 2.5g tank, IMO/E you need to make at least twice weekly water changes anyway for quality water for the fish and you will have it re-cycled in no time.....
To get your fish ready for the trip stop feeding him for at least 24-48 hours before you leave and make 50% daily water changes for at least 3 days before you leave, you want to try and clean him out so he doesn't poop a lot during the trip and cause the ammonia raise, it is also a good idea to have an ammonia lock product like Prime by SeaChem especially if you are going to be on the road for over 12 hours. And no food during the trip.
Have a safe trip.....