I wouldn't worry too much about the middle name, most people hardly ever use theirs on a day to day basis. Depends how strongly you feel about it though, if you really loathe it then don't, if just a bit indifferent to it, then why not, if it would make your mum happy?
Malcolm and David. Both plain and distinctly unfashionable, but they mean the world to me and my family so they stayed. But at least they're plain rather than truly horrendous, they're not names I imagine my sons will be so desperate to change that they'll go for a deed poll.
If I have a DS the poor child is going to end up with Bernard as one of his middle names for family reasons. Can't get more unfashionable than that! (I have decided it would be a step too far to inflict it on a DD)
Anyway, look at all the really unfashionable old-lady names that are now back in fashion. Olive, Pearl, Lily etc. No-one would have dreamed of calling their children those names in the 70s and 80s, now they are all the rage. So maybe your mum's name will be the Next Big Thing by the time your DD is old enough to care?
Agreeing with Ephiny regarding the insignificance of middle names. Literally no one outside my immediate family knows my middle name, although the initial crops up on official things occasionally. I likewise have no idea what middle names even very close friends and family have. And as other people have said, (un)fashionability is inherently transient.