Charging marine battery while powering equipment
There are many power supplies available for this purpose. As 0x6d64 says, these have a constant voltage output of around 2.3V per cell, ie 13.6V or 13.8V for a 12V battery. The power supply will need a current limit so that the battery will not be damaged if it is charged when flat. This would typically be about 0.1 of the capacity - so no more than 10A for a 100AH battery - probably 5A to be safe.

You can safely leave everything connected 24/7 with this arrangement and switch the mains on and off as necessary but some deep-discharge protection is advisable. This would cut the power output if the battery voltage drops below about 10V to prevent battery damage. Of course your connected equipment needs to be capable of operating at 13.8V for this to work. If 14V is the absolute maximum then this solution is borderline.