Yes the battery on the motherboard can cause issues if the battery has no charge...but usually the issues are not display related. This is technically called the "PRAM Battery". It would probably be a good idea to replace it in any case.
The PRAM battery probably looks like this:
View attachment 37209
If it were me...the first thing I would do is remove both video cards...then reinstall them. Just in case one or both of them are not installed/seated properly.
After doing this...if the same results are seen. If I was to guess...the video card that gives the black screen I would say might be bad...and the video card with the white screen might be good (but may not be progressing any further in the booting process...if there's no macOS installed on the hard drive).
Before we go any further...we should probably verify if the macOS is installed on the Powermac G4's internal hard drive. If it isn't...this could explain the issues.
Nick