Electronic – Battery/Charger Not Reaching Float with Load Attached

This is a valid and difficult question.

My suggestion would be

1. A DC-DC boost charger to charge the batteries from any DC source. If you're happy to just float them, try eBay for a switching PSU board.
2. Schottkey diodes from both Bat+ to Radio+ and PSU+ to Radio+.
3. A relay to prevent the battery from being used, at all, when the external supply is connected.
4. A "direct battery" connection point for your external charger.

Two diodes are needed to ensure that you don't back-power the supply, and finally, to make it possible to detect the external power failure, which you can't do if there's only one 13v bus.

The relay is needed because under full load the external supply will certainly droop below the voltage of a lead-acid on charge. A simple OR gate might not be what you want then.

The switch-mode charger might create a lot of RF noise... You can try to filter it...

Consider using two 6V batteries, and switching them with a relay: parallel for charging from an unknown source, series for supplying the load. Then you can use a linear charging circuit, current limited constant voltage.

Last thought - don't skimp on the cables and connectors. With 20A, every fraction of an Ohm counts. On my HF radio, the wire supplied with the radio suffered a 2 or 3 V drop when transmitting full power. Do the sums, go one gauge heavier than you think you need, and use decent connectors. This is where you'll lose the volts, not in the schottkey diodes.

Edit: seems I've designed the SuperPowerGate, and maybe improved it...

Make sure you over rate your battery capacity so that it cannot discharge to less than 60% of capacity after any expected power outage or have a under voltage cut out. Deep discharge is very destructive to consumer lead acid batteries.

Worst case figures:
Discharge margin 60%
Switching voltage regulator 80% efficiency
Low Battery voltage 10.2V (pick happy number)
Current required Max 6A (don't skimp here, it will not save money or much power)
Required 5.1V

Current required at 10.2V is 3A / 0.8 = 3.75 A Capacity for 15 hours is 56.25 Ah / 0.6 = 93.75 Ah

Your 100Ah battery should be up to the task if it is of a high quality and you do not cause it stress (high temp, deep discharge, over charge) or it will shorten life span.

You will need to charge at current high enough to fully replace over 60Ah in 9 hours of over 6.7A and the circuit consumption (worst case 3.75A) so at least 12A but better 15A at 13.8V because reaching final charge takes longer than a linear rating.

That would take about 2A from the 120V supply.

With unattended things a call out to check or replace a battery just once in 3 years may cost more than purchasing the next size bigger in all the components so this is what I would do and justify it just like that.

EDIT:
My calculations were fairly conservative except for the unknown recharge current required. I expect there will be tables and such to find out what peak current is needed to charge the batteries from 60% to 95% every night. Deep cycle are better and may be one of those over design ideas to implement, the 20% margin is what will save the day.
You may find that only 30% of the calculated capacity is used when everything is hooked up in the end (all the numbers were close to worst case). This will give you the safety margin for the reduction in battery capacity with age and the occasional unintended deep discharge due to roadworks, power failure, cable maintenance, human error.
Depending on the battery type you may have 10 to 100 deep discharge cycles before the battery is toast, try not to have any of these by design. It will be much cheaper to have a low voltage cut-out before deep discharge than having sporadic long power outages causing the batteries to degenerate too soon. Also do remember most all chemical cells perform worse when very hot (degeneration occurs) or very cold (capacity/stored charge vanishes).