Electronic – UPS 12V power switching

My thought would be to leave all 3 batteries connected to the UPS at all times plus add a 'proper' lead-acid charger as well. This charger would be a proper 2-stage or 3-stage charger with automatic float - equalization switching.

However, you need to be aware of a probable safety issue: the batteries are most likely LIVE with respect to ground when the UPS is connected to the incoming AC Mains. That's because there is NOT any isolation from the AC Main in most low-power UPS units.

That's one of the reasons that the battery terminals are so well insulated in small UPS units.

You can use an external charger but you must ensure that the charger can withstand being at a voltage significantly above ground and you must ensure that no one can touch the battery terminals while the system is live.

The best system is to simply have the charger constantly powering the battery. And the devices constantly drawing power from the battery at the same time. This is equivalent to a "full online" UPS. As there is zero transfer time. (cheap mains UPSes have a small drop in the mains output when they transfer the load from mains to battery during a power failure).

Check the output from your charger as mentioned above. As lots of 12V battery chargers have poor output voltage regulation. And are typically designed to charge a battery, and then be disconnected. They will often overcharge the battery if left connected to a battery 24/7. Get a regulated power supply that you can adjust the output to 13.5V or whatever the battery manufacturer specifies as the ideal "float" voltage for that battery.

Also consider how reliable your SEPIC power supplies are. Especially in relation to over voltage protection on their outputs. You don't want a faulty SEPIC inverter to destroy your connected devices from over voltage. Do a search for "shunt regulator" and "crowbar circuit" for examples of over voltage protection circuits.

And make sure that you put some fuses on the connections to your battery, as close to the positive terminal as practical. As lead acid batteries can provide very large currents if a short circuit occurs. The main purpose of the fuses in this case, is to stop a fire from occouring due to overloaded wires, if one of the SEPIC inverters or your battery charger fails short circuit.