If it were me, I probably wouldn't choose to use Freddie on the birth certificate. It 'might' open him up for teasing, and chances are at some point he will go by Fred anyway, so if you don't care for the shortened version, that's something to consider.
I'm having a similar problem with Theodore...I like Theo and Ted, but OH and I don't care for the longer name as much. But if we don't use the longer version, we have to choose one nickname or the other, we wouldn't be able to use both. It's hard, isn't it?
If you don't like Frederick, maybe use Alfred, Fernando, Wilfred, Winfred, Fritz, etc. My legal name is a nickname rather than a full name, and it's never bothered me, but mine also doesn't end in "ie." If it did, I may feel differently.