New thought: Does the splitter have some sort of valve to prevent backflow? If so, was it assembled (it came assembled, BTW) incorrectly.
As it came, hose 2 was the CO2 "In" from the regulator. Hoses 1 and 3 were "out" to the kegs.
I disconnected hose 1 from the splitter and connected it directly to the regulator and everything seems to work well. IMO this means the problem lies in the splitter.
I think I will try to put the CO2 "in" to where hose 1 currently is, and plug the kegs to hoses 2 and 3.
What do you think?