By no means an expert in horticulture but I do not believe that what you intend to do is
possible. Strawberry plants (at least those with which I am familiar) send out runners.
These will spontaneously start new plants and the problem generally becomes
thinning these out because a bed gets too crowded. They may be clipped and
transplanted but I'm not sure what is the best time for this. My guess, however, would
be after the parent has fruited. Then clip and transplant, disturbing the new root
system as little as is humanly possible and assuring that the transplant is watered well intiailly and continually thereafter until it gibves evidence of survivability.