I love Blythe. I vastly prefer the Blythe spelling. It's a gorgeous name. Blythe Genevieve is a lovely name. I also really love the suggestions of Blythe Alexandra, Blythe Magnolia, and Blythe Victoria.
A few other suggestions are
I would spell it Blythe.
Blithe is like blithering as in "blithering idiot".
I also like that Blythe goes up and down. When she spells her name there will be more of a dynamic and it's more interesting than if it never goes below the line.
Also Blithe would have to correct people on how to spell her name. Blythe wouldn't have to.
Lastly, Blithe contains lithe, which could be mean and strange to live up to.