Thank you for the suggestions! I'm wondering why only Canon's LC-E6/E charger is compatible with both... the one for LP-EL you sent has a meager 0,5A max with two bays used (vs 1,2A of the Canon charger, not dealbreaker but to be noted). The second one is very interesting with 1,5A×1 and 1A×2: this performance with the "flat" form factor and compatibility with LP-EL would be PERFECT.
I see that only the Canon charger has the additonal T and C contacts for temperature monitoring and other kind of communication, but both LP-E6P and LP-EL batteries charge at 8,4V, so I'm lost at why there are so few options out there. I guess all the wise people go Godox and live a simpler life...