For boy I like
Baxman (family name but I love the nickname [name]Bax[/name]) [name]Cole[/name]
Girl I like
Baeli (again family name pronounced like [name]Bailey[/name]) [name]Jane[/name] (I’m still not sure I want her initials to be BJ though)
I think that Baxman sounds a little too much like batman… what about just [name]Bax[/name]?
i think that i wouldn’t worry about BJ but go with the traditional spelling of bailey
Baxman (family name but I love the nickname [name]Bax[/name]) [name]Cole[/name]- [name]Way[/name] too much like Batman. What about [name]Baxley[/name]?
Girl I like
Baeli (again family name pronounced like [name]Bailey[/name]) [name]Jane[/name]- I wouldn’t worry about the initials, but agree with spelling it [name]Bailey[/name].
I think Baxman is great, esp. since it’s a family name. [name]Don[/name]'t worry about the Batman similarity - what little boy wouldn’t [name]LOVE[/name] to be called Batman! And when he’s older (which will be the majority of his life!) people won’t make that comparison.
[name]Ad[/name] I wholeheartedly agree with the [name]Bailey[/name] spelling. I would stumble over Baeli and want to say Bah-Eee-Li or Bah-L-ee
When I saw Baxman, I thought I was reading a typo of Batman, which isn’t a good thing. 
What about [name]Baxter[/name]?
I’d definitely go with the traditional [name]Bailey[/name], too, because while it’s a family name for you, most people will simply think you’re going for a “yooneek” spelling of the trendy [name]Bailey[/name]. On top of this, your daughter will have to deal with pronunciation and spelling issues.
Best wishes to you! 
I’m going to ditto the previous posters on two points:
- I read “Baxman” for “Batman” the first time around.
- I much, much, much prefer [name]Bailey[/name]. The other spelling will get butchered her whole life, and it just doesn’t look very substantial.
I’m thirding these points 
Also, I grew up with a [name]Bethany[/name] [name]Jade[/name], who went by BJ a lot of the time and she really didn’t have much trouble at all. Nobody picks up on the initials until high school, and even then the only time anyone mentioned it was when one of her friends wanted to pay her out for something, so I wouldn’t worry too much!
I agree with moxie, the batman comparison isn’t necessarily a bad thing. I can think of a lot worse things to be called as a child. [name]Cole[/name] would be after my sister [name]Nicole[/name]. I would go with a completely different name before going with [name]Baxley[/name] or [name]Baxter[/name] since the whole point was that it is a family name. Although I do really love [name]Baxter[/name] I think I know too many dogs with that name.
I really didn’t expect the hatred of the spelling of Baeli. It’s phonetic really bae-li. Again it’s my grandmother’s maiden name which I have always loved and I would go with a completely different name before changing the spelling. (Which is still a possibility)
I am a traditional name lover, and generally don’t prefer the last name as a first name trend, but I actually love Baxman [name]Cole[/name]! [name]Bax[/name] is a great nickname.
I don’t, however, like Baeli. I have no problem pronouncing it, but folks are sure to assume you just liked the name [name]Bailey[/name] but wanted to spell it in a unique way. Also, I just don’t generally like the name, even spelled traditionally. That being said, I think Baeli as a middle would be very nice. Flow-wise Baeli in the middle is a little tough, but I think if you love the name and it has meaning for you, that is all that matters. Would you consider [name]Jane[/name] Baeli?
Some other thoughts for girl names that would go well with [name]Bax[/name]:
[name]Catherine[/name] Baeli ([name]Cate[/name] and [name]Bax[/name] are so cute together!)
[name]Sarah[/name] Baeli
[name]Olivia[/name] Baeli
[name]Lillian[/name] Baeli
[name]Gwendolyn[/name] Baeli
[name]Sylvia[/name] Baeli
Good luck!