In the name of this band, The is part of the name and ought to be capitalized. Also, since the song belongs to the band, an apostrophe should be added to the end of Beatles to make it possessive.
Titles of songs should be always written inside double quotation marks.
Don’t capitalize with; it’s one of those “insignificant words” that doesn’t get capitalized in titles.
The last sentence of this example should read, Scientists couldn’t resist naming it after The Beatles’ song “Lucy in the Sky with Diamonds.”