If they were independent in 63 and had a Constitution in-place that has not changed that much except amendments according to their site, I don't know what else they would be called.
It sounds like the difference was that in 64, they declared themselves a Sovereign Republic.
Yes, they had a constitution and that constitution was amended so that Kenya became the Republic of Kenya on 12/13/64. If anyone can show official Kenyan documents that are letter-headed Republic of Kenya prior to this date then this is a non-issue. Until then it is a hole in this theory.