Yes, PLEASE get a better battery to use!! Really, besides a small subset of vapers who only use low wattage and know exactly what they're doing, anyone planning on using the full features of a single-battery, 75W, TC-capable mod should be using a battery with at least 20A continuous discharge, even a tad more if you're willing to sacrifice some runtime (but 20A should basically be sufficient).
Your 10A battery is probably a big part of the heat issues, too.
Although just so everyone knows, during PBusardo's torture testing, he did find that the mod has an error reading/shuts off when the internal temperature gets too high.
Typically, I use green Samsung 25Rs with my Pico. I also use VTC4s from IMRBatteries, but I would hate to suggest those and have someone end up with counterfeits. Samsung 25Rs are good batteries.
Lastly, anyone in the market for a new tank and wanting one for the Pico specifically, I'm absolutely loving the Lemo 3. I am so impressed. It definitely has some kind of weird quirks about it but BOY does it perform. I'll take the quirks. The airflow might be a bit much for some but yeah, really nice tank. It's also great that it takes the same coils as the Melo, not to mention has a really solid RTA deck.