Look all, take this for what it is but I happen to come across this youtube video. The presenter is somewhat eccentric but he explains in a little detail why a poor battery could cause a drop in MPG. This was completely coincidental but seems to confirm the the information I had gathered from other sources.
Again remember, eccentric - youtube - internet, so don't shoot (me) the messenger. This is all for purposes of civil discourse and of course to help the OP.
https://youtu.be/rEDkdPJHScA?t=165