Can a stomach bacteria be spread through saliva on lips?

Probably: Certainly, helicobacter pylori transmits from person to person and oral-oral transmission likely occurs as one route of spread.

Yes: Bacteria are plentiful throughout the gastro-intestinal tract the GI tract begins at the lips of the mouth and ends at the anus. Some of the 'normal flora' of the GI tract can be found throughout the system, some of the bacteria are more regional due to other factors...Ph, oxygen concentration, etc.