Not certain it is the same Mitchell Lace, however, a Mitchell Lace (1642 to 1700)was the son of Donold Lace and Jony Kaighan. Donold was a farmer at Kerrowdhoo in Bride, the Lace family having been on this farm from at least 1515. Their son Mitchell married Alice Christian and began to farm in Andreas.
This Mitchel also had a son named Mitchel, so that may be the one you mention. He would have been born in Andreas and about the correct age.