Re: inverter- alternator size
Since you are staying under the continuous duty 20 amp rating on the AC side (2500 watts/125 volts) let's look at something else. Your DC and AC current readings don't jive. Watts equals amps times volts. To find the 12-volt DC draw required, multiply the AC amps by about 11 to include inverter efficiency losses. When the AC is drawing 19.8 amps, the inverter is drawing about 218 amps. When the DC is drawing 44 to 46 amps, the AC could only put out about 4 amps.
My guess is that you did not make the readings at the same moment, and the refrigerator or air conditioner cycled off between the AC and DC readings.
So, not seeing your system, I have two guesses about the shutdown:
1. You are drawing more current than the alternator can replace, and a low-voltage disconnect in the inverter is shutting down to save the batteries, or
2. There is not enough cooling where the inverter is mounted, and a temperature sensor is shutting down the inverter to protect the solid-state devices.
Either of these could also be made worse by DC wiring that is too small, too long, or poorly connected.
[geek speak warning]
Lets say you are running 2500 watts of AC loads and have 90% inverter efficiency. While charging at 14 volts, you will draw 2500/14 or 178.6 amps to power the AC, allowing for the losses in the inverter figure that the total DC draw would be 198.4 amps. This is equal to a resistor that is 0.07 (7/100) of an ohm. If I add 0.03 ohm of added loss in the cabling, the voltage at the inverter is now only 70% of the battery voltage, or in this case 9.8 volts. But many inverters will try harder, and attempt to draw 283.4 amps at 9.8 volts to continue supplying the requested 2500 watts AC. This causes the voltage to drop even more, and the current goes up more if it can. Meanwhile, the "lost" voltage is actually heating high resistance connections, and could start a fire.
So, the lower voltage due to an "almost good enough" DC wiring draws ~ 280 amps instead of ~ 200 amps, increasing heating of the solid-state devices, and causing either an early low voltage or a thermal shut-down.
[/geek speak warning]
So the best practice is:
1. Mount high-current devices like inverters very near the batteries, and keep the wires short.
2. Use the heaviest wire possible, without being too big to fit the connectors securely, or cause mechanical stress on them.
3. Make sure all connections are clean and tight. "Good enough" isn't good enough with high current.
4. Make sure the inverter can get a reasonable amount of air circulation, and any fins or fan ports are not blocked.
You may want to wire a voltmeter to your batteries, and not look at a charge/discharge ammeter only. See if the battery voltage is coming up when charging, (13.2 to 15 volts) or instead just breaking even (12-13 volts). You also may want to compare the voltage at the inverter with the voltage at the battery under full AC load to see if there are serious (more than ~ 0.3 volt) voltage drops.
Someone said "Making good decisions comes from experience, experience comes from bad decisions." I say there are three kinds of people: those who learn from their mistakes, those who learn from the mistakes of others, and those who never learn.