its safe, solar charge controller wont be damage by battery or alternator. If the controller sees a full battery it goes into float mode.
The battery is more powerful than either the alternator or solar charge controller, it is capable of releasing more amps in one second than any charging system can provide. You can connect as many chargers as you want to it, the chargers will only see a 12 volt battery and if it needs charging they make amps available to it.
Once a 12 volt battery is full it wont accept any more amps, even if your alternator puts out 200 amps, the battery wont accept it. You can keep the battery connected 24/7 and it wont be damaged.
The only thing that will damage your battery is if the alternator is putting out more than 15 volts. Volts is what forces the amps into the battery, most chargers/alternators max out at 14.4 volts, this is enough to charge a battery without overcharging. But at 15 volts or higher it will force the amps into a full battery and it will vent.
As an example the 12 volt jump packs have a 17ah agm battery, those can be charged from a cigarette lighter (male to male plug) and I have connected them to my 240 watt solar panel without any problems, and that is one of the smallest agm battery.
The inverter has to be connected directly to the battery, hardwired is the best way.