1- I would take the LH from FRA as it gives you an extra hour to clear immigration and customs. There´s a Fast Sleep Hotel that is landside on Terminal 2 where you can take a quick shower.
2- A suggestion would be to take a taxi to Boizao Grill on Marginal do Tiete already closer to GRU. Enjoy a nice rodizio lunch and make your way from there to GRU by taxi.