Well, I'll chime in with some advice:
Are you SURE that if you move LR over from your big tank, no pests will come with it? If there's any chance something might come, I'd start the tank with some dead live rock (oh, THAT makes sense): Put some live rock into fresh water for a week or so. Then move it over to the new tank. Then cycle normally.
If you are SURE of your live rock, then bring it over. No reason to use old tank water, unless you just want to.
If your big tank's water quality is good, you can change 10-20 gallons of it weekly or every other week, and use it in your frag tank.
Or:
Salt is cheap. In a 20 gallon tank, change 50% twice a month, or 25% once a week. That's 5 gallons a week. A $50 bucket of salt will last you 6 months. You should be fine.
Bill
215g FOWLR... and anemones, GSP, gorgonians... carp, that isn't FO!
"I killed my first SW Fish in 1971..."