Bartholomew is an alcoholic. He was constantly seen drinking from a beer mug and became disappointed whenever he ran out. During Ratigan's song, when Ratigan dumped a bottle of champagne into his water fountain, Bartholomew got very excited and immediately ran over to the fountain, where he started drinking from it. Then, Ratigan kicked him, and Bartholomew fell in and started swimming around in it. When Ratigan talks about his history with Basil, Bartholomew started crying (which was most likely because of his being drunk).
Later, when they are finishing up a verse of the song, he drunkenly says, "To Ratigan, the world's greatest rat!", which makes Ratigan very mad (as he despises being called a rat). He threw him outside the hideout and summoned Felicia with a bell. However, Bartholomew was too drunk to notice her, and she easily caught and ate him. After a short period of mourning from the other henchmen, Ratigan suggested that they finish the song before he had Felicia eat the rest of them, which they do out of fright.
It is possible that Bartholomew was still alive inside Felicia's stomach; when she eats him, she merely swallows him, as there was only a gulping sound heard rather than a chomping noise. Nonetheless, he would have probably been digested by the end of the film, making this a moot point.
However, because the gulping sound came before the other henchman closed their eyes from seeing Felicia eat Bartholomew, its possible that the gulping noise came from the henchman.