If he is going to be compared to a Lincoln, I guess Abraham Lincoln makes a better analogy than Benjamin Lincoln, since Abe was elected to the House of Representatives but was defeated when he tried to win a Senate seat. Benjamin Lincoln lost Charleston—if Beto is the nominee in 2020 he would probably carry Charleston (Obama carried it both times).
No, Abe left the House in 1849; he did not run for the Senate against Stephen Douglas until 1858.