Hello, Hypers.
It sounds like you are using the #4022 Romi Robot Kit for FIRST with the WPILib software. If so, please note that we are not involved with the software and Raspberry Pi side of the WPILib project, so you would need to contact the WPILib team for problems like the wireless connectivity (although it sounds like it could just be a poor WiFi signal).
As far as the batteries, I suspect the Raspberry Pi is a majority of the current draw, so you could look up ways to reduce the current requirements of the Raspberry Pi to see if any of them would be practical for your application. For example, you might try some of the suggestions in this write-up on optimizing Raspberry Pi power consumption. Additionally, if you are using Alkaline batteries, you might try switching to rechargable NiMH batteries instead.
The video you are referring to uses an RC receiver to control the servos for the robot arm. You can use the Romi 32U4 Control Board to control the servos in the Robot Arm Kit for Romi, but unfortunately, the on-board regulator cannot handle the additional current draw, so we recommend adding a separate external regulator, such as the D24V22F6, to power them. We do not have any specific examples for doing this, but you can find some details (and other considerations) in my post in this thread.
Brandon