My son is 24 now and is Edward on his birth certificate, but has always been known as Ted. We called him Teddy a lot when he was small, and still sometimes do playfully, but he has always been Ted. He doesn't really like the Edward on his birth certificate and never uses it except on CVs etc. It's slightly confusing for things like bank accounts and tickets as the initial E is different from the T he uses normally.
When he was in nursery and primary school, we only knew one other Ted (and he was Theodore on his birth certificate), but the name became a lot more popular a few years after he was born.
No one has ever teased him about soft toys or having a childish name. It's a long established name. There have been loads of well-known adult men called Ted and Teddy during history - including Teddy Roosevelt, after whom teddy bears are named, Teddy Kennedy, Teddy Sheringham, Ted Heath, Ted Hughes (the ex poet laureate), Ted Bundy (ok, not a great association), Ted Danson (the actor in Cheers), Ted Lasso, Ted the love interest vet in Shits Creek, my great uncle Ted ...
I say go for it, but I would put another name on the birth certificate as I'm a traditionalist.