Good on you for moving them to a new tank. That should give them more room. I did a similar move quite early in my fishkeeping days, and the following is a combination of what I did and things I would do differently now I have more knowledge.
You have exactly the right idea in keeping the water and components the same. That should minimise the impact of the change. Keeping the same water should minimise the stress to the fish and keeping the same filter should keep the bacterial consistency of the filter medium the same.
The biggest problem I see with your plan is the substrate: there's usually a lot of crap on the tank floor, and it might introduce more problems than it would solve to try to migrate the gravel without the crap. As soon as you start to try and shovel it up, all that crap will start floating around in the water. If it were me, I'd probably use the opportunity to buy new gravel. Rinse it thoroughly before adding it to the new tank to remove dust, soak it for 24 hours in dechlorinated water (tank water if you have any to spare), discard the soaking water and then add the gravel to the new tank. In the old tank, suck the water up until about an inch off the floor (point the tube at the edge rather than the bottom of the tank, to avoid sucking up the crap) to go into the new tank, then ditch the last inch's worth of water and the old gravel.
Alternatively, to keep the gravel, I'd say suck the water up as above, leaving the gravel undisturbed for as long as possible, then take the gravel out, put it in the sink and rinse it thoroughly to get rid of all the debris and dirt. (Ideally you'd rinse it with tank water, but it'll probably need quite a bit of a rinse, and the water you use to rinse won't be useable for the new tank since it will be full of crap. On balance I think it's better to rinse with tap water and have more water to put in the new tank.) Then add it to the new tank and continue from there. This might be a better option than replacing the gravel entirely, since you'll lose the crap but still have any algae etc. that are on the old gravel, and don't risk altering the pH or introducing new solutes into the water with the new gravel.
You've got the right idea with the water changes as well. Too much of a change will stress the fish, and fresh water from the tap is completely different from water they've been living in for a while. Starting off with the minimum you need to allow the filter to work is a good idea - they're going to be stressed enough from the move anyway without a huge water change. I'd leave it at least a week before adding more, but also test the nitrate/nitrite and pH before and after moving, to see if anything's changed.
As long as the tests are okay, I'd consider adding 20% more at a time, as if it's just a normal water change. So after the move you'll have 11.5 gallons. You could add 2.3 gallons in the first water addition, then 2.8, then 3.4, which would take you up to 20 gallons. If you want to stay on the safe side, you could reduce those a bit so it would take you one more change to get up to 20.
Of course, if you get an ammonia or nitrite spike, you'll probably want to change some water as well as adding some. So you could take out 10% and put in 20%. You'll get to 20 gallons more slowly that way but it will enable you to remove some water as well as just adding.
Finally, I'd recommend leaving the filter cartridge in there for a bit longer so as not to disrupt the filtration system (in terms of bacteria and so on) too much. Alternatively, or additionally, rinse it at your next "change" instead of discarding it entirely. When I kept fish I had a filter which had two spaces to put the medium, which allowed me to change only half at a time, which was great for keeping the bacteria consistent. I'm not sure from your description that you have one like that but rinsing it would have a less dramatic effect than binning it and starting afresh.