Line breeding is perfectly acceptable in chickens and WON'T cause fertility or deformity issues until many generations of close inbreeding between brother and sister. Not father to daughter or mother to son. That's actually how you avoid issues. Many people including myself, "line breed" and keep a CLOSED flock successfully without EVER bringing in new blood.There are reasons we humans have laws about incest. By accident we bred full brother to sister and got only 1 egg to hatch.
Chickens along with many other species don't pop up with inbreeding issues the way humans do, so fast. Maybe your birds were already from generations of CLOSE inbreeding? (no ugliness intended at all)
We as humans have MORAL issues with incest. Animals DON'T have these same morals... and have been breeding for longer than humans have existed.
Here is a link to an easy to follow program. Could even separate the hens for individual mating and start individual pens... to expand on the current genetic diversity with just the 3 birds... yes it's much better if your first generation is not full brother and sister... And EXTREMELY important to select for vigor...
http://www.maransofamericaclub.com/inbreeding-chickens.html
But seems like a vegan (no ugliness intended, please just an open discussion about chicken flock management)... might have a harder time "selecting" for anything.

And I know the OP hasn't given any indication of hatching intentions, thus far... but just as part of that suggestion... what would be done with all the extra cockerels? I do keep a stag pen (to control over mating, and genetics) and If you have the means to support and can tolerate all the extra crowing, boys CAN get along forever even if they weren't raised together. Or not, sometimes it varies. Is there an issue with letting another family harvest for their table? We do eat our extra Silkie boys (Sorry OP, I really don't mean to make you uncomfortable).
Also note that Silkie moms are NOT friendly or pet like and will do their best to teach their offspring to avoid you. But they also teach them to come when you have treats. And once they grow, all seem to be individuals as far as flightiness verses friendliness. And yes, broody's with babies are a lovely sight to behold!
