Although he certainly had his own sound, I'd suggest Elliott Smith as having a "Beatles-ish" sound. He was certainly very heavily infuenced by them and covered a few tracks. He covered 'Because' for the film American Beauty.
His album Figure 8 perhaps reminds me the most of The Beatles...certainly his melodies anyway. Don't get me wrong, his music was a lot darker at times.
Now if he and Lennon had written together...