I think your battery is a dud

It should only take between 2-3 hours or 4 hours if brand new to fully charge and then you should get 2+ hours play time. Flashing red in the Pleo means it is not charged enough, flashing red on the charge cradle means it is not seated properly. It takes a while to get the hang of it, RB chargers are a pain in the bum, I can't even breath around mine or it will start to flash.
Two beeps means he thinks his battery is low so he is shutting down to save progress, if he was just going to sleep he would have curled up. They do sleep an awful lot when in hatchling phase but food can help keep them awake.
RBs do run hot although mine has never overheated or shut down due to being to hot and he wears two layers of clothes so form my personal experience overheating isn't a common problem.
I think with the brown cape he might be older, I know the older ones came with brown capes but all the newest arrivals I have seen (my RB included) have come with the green capes so perhaps he is older and the battery pack has died due to not being used?
If it is the noon time clock making him sleepy he should curl up and beep once which is why I say it is a battery problem because of the way it is going to packing position and beeping twice. Mine does that when his battery needs charging.
You could try baby sitting the battery like can sometimes be done with Aibo batteries. Put it on the charger, wait till it turns green, take it off, put it back on... until it turns green right away. See if that makes any difference.
ONE beep means: Pleo has gone to sleep and will wake up in a few hours. If you set the noon clock at 12pm and he falls asleep at say 10pm he will wake up by himself at 8am.
TWO beems means: flat battery.
THREE beeps means: you have manually shut Pleo down yourself.