We double barrelled our kids names as I didn't take husband's name, quite deliberately. Paternalistic society point, and not wanting to continue what largely started out as the passing of women as chattels (from father to husband) on marriage but also because of work - my name was my reputation.
It sounded shit at first, and people took the piss, but it's THEIR name. So it works. They are a combination of the two of you and should have both names. Who gives a fuck what anyone else thinks? They love having both names and complain if anyone misses part out accidentally.
Out of interest, many of the Children in my kids classes have double barrels for the same reasons, either because their parents aren't married or don't share a name. Some flow at first blush, some don't, but no one bats an eyelid. It's very normal these days.