Giving your project a little more thought (regarding my self charging comments earlier as an alternative to bigger, better batteries) while walking the dog today:
1). Your charge contacts can only be located on the bottom of the HEXA Body as the Head Rotates and the Legs present obstacles to overcome (this is probably why the designer used a round charger base for inductive battery charging in the first place).
2). Rather than a challenging 2 contact pad arrangement as a mating contact target on the charging base, I would suggest thinking about a Brass Disc in the center of a Brass Ring (think very simple bullseye) for the charger base and 2 contact pads on the bottom of the HEXA (one inner and one outer matching the "bullseye dimensions) that would make contact with the Base Contacts regardless of the HEXA orientation when seated.
You may inquire to Vincross if they have any of the discontinued charger discs as these are a perfect fit to the round HEXA bottom (the Disc has a mating indentation). If they can’t supply one for your retrofit, they may have a 3D file for printing. At least the dimensions could help so that you could fabricate something. In a pinch, I have one and could provide you measurements.
You may have noticed that HEXA always comes to rest with its bottom contacting the surface of the floor, this programmed at rest position could be utilized for this self charging by electrical contact purpose (it was probably built-in for the original inductive charging disc use).
The original HEXA project carried a plant on its head and sought out light sources. I believe it also sought out a self charging base (may have to confirm with Vincross). If I am correct, perhaps the required code has already been written and all you need to do is adapt it to your project (if someone at Vincross has access to this and is willing to assist). Except for the physical charging method (electrical contact vs. induction by close proximity) all code instructions should be exactly the same or very close.
Although this electrical contact method is not as elegant and esthetically pleasing as the inductive, Vincross could consider using this approach with future HEXAs as the self charging feature was originally intended. It would extend use time especially in high drain applications. HEXA reaches 20% batter level and the App asks if you want to send HEXA to its Charging Disc (or a completely self directed automatic function).
Again, Good Luck with your project!