Will the Summertime Mistress card be a drop in part 2?

No, instead, there's Chaldea Lifesavers on part 2 shop+very rare drop: https://grandorder.gamepress.gg/craft-essences/chaldea-lifesavers

1 HP Guts+NP generation on 15%

Summertime Mistress also gives no event bonus on part 2. All the others from part 1 do, on the other hand (Shiny Goddess, Pirate's Party, Twilight Memory give bonus to the new, but different mats).