These guys will breed every 2-3 weeks so you have plenty of time to practice. Here are a couple of answers:
1) In order to cycle your new 10-15 gallon quickly, add water from your existing 60 gallon (as you already have), take some used filter material from your main tank and place it in filter for the new setup, and then take at least 5-10 pounds of live rock from your main tank and put it in your grow out tank. This will keep the water quality up and the rock/filter material will have all the biological organisms necessary for filtration.
2) If you cannot move the rock they laid on, then as people have mentioned, wait until they hatch and take a turkey baster and carefully siphon them out to your new tank. If you're worried about other fish eating eggs, leave the lights on 24hours a day to keep your clowns awake and active. They will protect their young. This is an old trick used when breeding discus.
3) Lastly, consider setting up a separate breeding tank...like a 20 gallon and move some of your live rock and anemone into that tank. They are extremely territorial of their anemones and are used to moving around to follow their host anemone. As long as their host anemone is present, there is a good chance they'll feel comfortable enough to begin breeding again in the new setup.
4) Live rotifers are best but as an emergency backup, you could use some liquid coral/invertebrate food mixes that are commonly available...they are rich in rotifers, phytoplankton, and zooplankton.
Good Luck!