I'm very sorry, it looks like I just forgot or didn't understand how a mosfet gate works^^ I'm so ashamed. The whole time I had in my head to limit the current at the gate by means of a resistor, but that's total nonsense, unless it is provided by a microcontroller, that needs to be protected^^ Oh man! The gate is just a capacitor that should charge up as quickly as possible and that's it, not a continuous current flow.
My circuit "works", but no matter how I install the resistors, it will always only switch on hard or switch off hard, never both. At least I don't see any way of doing this (with one transistor), because I need the resistor in one place to avoid a short circuit. Now I finally understand why it shouldn't be done like this! That’s a total fuss. I feel like I had a huge resistance in my head that is now gone. All because I had a wrong understanding. I thank you for being so patient with me. Now I know what to do.
Since I couldn't find a driver for my voltage level, I will look for a PNP/NPN Transistor in one SOT package. I think I saw this somewhere. Just need to find one suitable one for an 1S Battery.