Discl.: This is not a Surround answer, just (three into) one that give(s) the result your are expecting -- note: there is a big limitation, Surround can work with any tag, here you'll have to known your tags in advance (well, lh-brackets could ask for the tag to insert, but that wouldn't be very ergonomic) .
With lh-brackets, it'll be solved with:
:Brackets <div> </div> -nl -insert=0 -trigger=µ
Then you select your lines and hit
µ. The magic is in the
Otherwise, you have the old way (define your own mapping) ... As I though, I've answered to a very similar question not so long ago, but on SO.
:xnoremap µ S<div><cr><c-r>"</div><esc>
" select your lines, hit µ, and then, you'll need to reindent
xnoremap µ <esc>`>o<div><esc>`<O</div><esc>