This is what worked for me - although on a much larger tank. But I still think it'd work.
Siphon good water from the 20g and prepare it for transport. Get about half of it. With the remaining half, stir up the sand bed and siphon that water along with a lot of the muck. Keep stirring and siphoning. Remove about half the sand and take that along also. Of course this is assuming you've got some new mixed SW
ready to go.
When you set up, you can use the 10g of old water, half the sand, new sand, and new water with the rock you have. Of course there'll be a bit of a dust storm for a few hours.
That old sand is good bacteria laden/seeded and I'd hate to see you ditch it all. No new cycle, but still watch your levels before you add the critters back.
After all is settled, do a PWC
That's my 2 cents.