I don’t think the Dems would fight to give Obama a third term as it means that Republicans would also get a chance at a third term.
But, he wouldn’t be the first president that hoofed it back to the Senate after leaving the presidency.
Or even scarier- pull a Taft and convince the next Dem president to name him to SCOTUS
Actually, I don't think any president, out of office, has gone to the Senate. John Quincy Adams went to the House of Representatives where he fought tooth and nail for 10 years to exclude Texas from the Union permanently.