Hmm, I'm honestly not sure. Obviously putting the rock with algae on it will bring the algae with it. But perhaps if your parameters are cleaner than the last tank you could get rid of it? Maybe try scrubbing it off gently before transferring?
Is there any way you could trade your live rock for other live rock? I think that would be your best plan. You wouldn't have to worry about transferring anything and you should avoid an all out cycle. Because IMO
letting the tank run for one week with biospira won't be enough to call it cycled.