IIf we speak about competetive god to Lolth - it will be Vhaeraun (god of drow males, thievery, territory, shadow magic, spellfilchers, and evil activities on the surface, aimed to further drow goals, interests, and power there) nothing will change except that this god attracts drow who want equal rights in gender part of society - overthrow matriarchy.
Oh, also, funny you don't bring up Eilistraee when talking about competitive Gods to Lolth. I heard there is a recent change to drow society about multiple factions emerging which allows for more alignments, but I'm afraid I don't know enough to speak more on the matter. Something interesting to consider I guess! I don't think it's
entirely impossible to reshape it.