Which One's the Nickname?
Okay, MS, this is an easy one. You are right and your husband is wrong. Yup, that's what I said: you are right and your husband is wrong. Go ahead, take your victory lap, then meet me back here and I'll explain.
Ready?
Your husband is right that naming conventions are looser than they once were, and it's well accepted for a baby's full "formal" name to be a traditional nickname, like Charlie instead of Charles. And he's also right in that some nicknames are longer than their formal sources. For instance, Nancy can come from the given name Ann, and Angelita from Angela.
However, in the case of Bea, nickname Beatrice, you have a traditional formal name and a traditional nickname for that formal name -- you've just flipped the normal roles. And names aren't naturally reversible that way. They shorten a lot easier than they lengthen.
Think of it in terms of clothing. The longer name is like a long-sleeved shirt: you can wear the sleeves long, or you can roll them up. But the reverse isn't true. If you start out with short sleeves, there's no way to lengthen them later when a breeze picks up.
What's more, reversing the formal name and nickname crosses up people's expectations. That will no doubt create a lifetime of little name aggravations for your daughter. To what purpose?
If you and your husband decide you love Bea but not Beatrice, name her Bea and call her that. If not, go with the more formal Beatrice, knowing you can call her Beatrice or Bea or even one of Beatrice's other nicknames. If your daughter wants to adopt an alter ego someday, Bess or Bebe or Tricia or even Trixie might fit the bill.

Comments
Or Bix.
I second the "you can shorten but it's hard to lengthen" argument. You can nickname kids whatever you want in the comfort of your own home, but it's less socially accepted, for example, for someone legally named "Kate" to "become" a Katherine or Kathleen in the public world, than it is for either of those to take a shorter nickname. And people who are given nicknames as names tend to get miffed when people assume they were given the longer name and they're "just" going by a nickname.
I definitely agree, you can't get a longer name out of its nickname. I know a girl who is just Kate and she gets Katherine all the time and it bugs her a lot. Or my friend's sister Eliza gets called Elizabeth even though Eliza's been recognized as a name on its own for a while now. It's silly to try to get the longer name from the nickname first name.
Bea as a name is really cute but I would go with Beatrice. (Or Beatrix).
First of all, I love the sleeve analogy!
And I agree, it's too confusing to make a longer name out of it's nn.
Also, I forsee potential issues with choosing Bea for legal documents and such, as it's more natural for one to assume that Bea is a nn for Beatrice.
When the latest baby name stats for the UK came out, with all those cutesie nicknames-as-names, I started wondering: twenty years from now, when all these Alfies and Jacks are grown up, will it become normal to flip the normal roles of full name versus nickname? That is, will the current questioner's husband become correct in the end, because all those nickname-only young people will want options?
In the meantime, please give the child a full name. Beatrice or Beatrix both work nicely.
If he doesn't like Beatrice or Beatrix , there's also the lovely Polish name Beata (bay-AH-tah) which can also be shortened to Bea. Means Blessed in Polish.
I have heard of a short name being lengthened in the instance of one name: Luke. I've known several people with the given name Luke who get called Lucas as a nickname to the extent that you might think Lucas is the actual full name. But that's it.
(I've also run across someone with the given name Johnny who goes by John, in a reversal of the usual nickname/full name set, although it doesn't really apply to the OP since Bea is a nickname that's shorter than Beatrice.)
I would also recommend the full name. My name is Lori, which for a while now has been a full name by itself. But prior to the late 20th century it was typically a nickname for Laura, Lorraine, or Lauren. So every once in a while an older person asks me if my name is short for something, or (annoyingly) assumes that it is when it isn't.
I have to agree with everyone here, please give her the longer name otherwise she will go through life correcting people and despising the longer name. My legal name is Jenny and everyone automatically assumes its Jennifer. And it infuriates me, especially when I introduce myself as Jen that someone would then start to call me another name.
Anyway my point is, give her the longer formal name and then shorten it :-)
My name is Mike but people constantly refer to me as Michael, which really annoys me. I introduce myself as Mike but for some reason many people decide to edit to what they perceive is my given name. In a way I find it arrogant.
My brother is a Ben, my Dad didn't see the point in naming him Benjamin when the "jamin" was never going to be used. People constantly assume he's a Benjamin and my brother hates that he's just a Ben, and wishes that he had the full name.
Beatrice is such a pretty name, anyway!
Cabo: beata and beatus are Latin adjectives (female/male, respectively) meaning "blessed". (Hence the English words "beatify" and "beatitudes".) Lots of languages have borrowed it as a name, including Polish (and Hungarian). It's not much used in English, probably because of the unfortunate written resemblance to the word "beat", as in "beat up".
I have a friend whose husband goes by Ted - as a "nickname" for Teddy.
He puts it this way; "I don't know what my mom was thinking..."
I think the longer name as the formal name is better, but what the child is ultimately called is hopefully going to be driven by her. I knew a girl named Jennifer who ended up "Jen" & one called "Katie" who (at the age of 4) decided she was too grown-up for that and became Katherine. (The longer names were their formal names.) Unfortunately, people are always going to do their own thing & call you what they want. In the instance of my Comadre (god-daughter's mother), her birth name is Christy, she goes by "Chris", but she is often referred to as "Cristina". Talk about confusing...
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