AzertyBot

That was it, now working perfectly. Thank you very much for the fast answer !

KSP : 1.04 Windows 32-bits Mods : Reproduction steps : Place a command pod/probe (any), a battery (any, with <100% Electric Charge) and a solar panel (any). Launch at KSC during day, extend solar panel if needed, watch batteries not recharching despite solar panels getting sun exposure. I tried with the rocket that failed, then with a OKTO + battery + solar panel, then I tried to replace the OKTO with a MK1 Command pod : no success.

I remember doing this mission, took me a bunch of minmus mining/fuel transport to accomplish For stations i strongly recommand you to put a science lab in it (with experiments) as it's a huge science generator. Don't forget to equip your station with solar panels (put more solar panels the farther you are from the sun as their effectiveness decreases with the distance). A fast antena and a lot of battery will help you too, as the science transmit takes time and a lot of energy. A probe core somewhere could also be a good idea, in case you need your kerbals somewhere else. As others said : docking ports are almost mandatory.

RAPIERs are fine : they make SSTO really easy to make, but given their science cost it's well balanced IMO. Maybe more heat issues ? I don't know how SABRE engines works in real life, but in KSP i never experienced this much difficulty with RAPIER and overheat, and it would make a use for engine coolers.