LOCtronicz
Formerly known as zenloc
What you want to do is simple but does require a 3 way switch or a power port with sense switch.
I have that internal charger in one of my projets and have the batteries wired to that and my 3 way switch. The power of my adapter to the other end of my three way switch and the internal charger. Now when I turn on the switch to battery is plays of the batteries and when I switch it to powerport I play the cube from the powerport but at the same time it turns on the internal charger and chargers my batteries while I'm playing. The batteries are therefore not being discharged. It takes around 1.2A to charge and the cube takes about 1.7A + 0.5A for you screen and sound amp (can me more or less depending on your setup) you need a 3.5A power brick to play and charge.
I have that internal charger in one of my projets and have the batteries wired to that and my 3 way switch. The power of my adapter to the other end of my three way switch and the internal charger. Now when I turn on the switch to battery is plays of the batteries and when I switch it to powerport I play the cube from the powerport but at the same time it turns on the internal charger and chargers my batteries while I'm playing. The batteries are therefore not being discharged. It takes around 1.2A to charge and the cube takes about 1.7A + 0.5A for you screen and sound amp (can me more or less depending on your setup) you need a 3.5A power brick to play and charge.