Tthe  passage  on  "little words" in Hebraica Cataloging (p. 21) says:  "In the rare cases where Even-Shoshan may
show the  same word under more than  one entry element, the  more  'analytical' option is chosen."

In Even-Shoshan under "lamed" there is an  entry for: " lamed (sheva) bet (patah) dalet"
under  "bet" there is the following:  "lamed (sheva)  bet  (patah) dalet,  bet  (hirek) lamed (sheva) bet (patah) dalet,  mem  (hirek) lamed (sheva)  bet (patah) dalet, see lamed (sheva) bet (patah) dalet."

Therefore, this is *not* a case where Even-Shoshan shows the same word under more than  one entry element.  He shows the  word under lamed, and  under  bet  he refers the reader to  the  lamed entry.  In short, he treats this compound as  a  "word" beginning with lamed.  Therefore again, there's no "more 'analytical'" option to be chosen-- he doesn't give  an  option  at  all.   So:   levadi  is correct.  Even-Shoshan does say, in  the "levad" entry, that the  word comes from "le-" plus "bad."  But it's the  fact that  he doesn't give an *entry* for it under "bad" that's critical.