Even a small load will affect a low soc battery. So drawing down the battery, then removing one of several loads, and still getting the alarm is an indicator of low soc / Vbatt.
I can run my 700w microwave (1100w+ draw) on my 1000w inverter, no problem... provided I have the battery capacity to back it up. The only time my alarm has sounded was on half the bank (a single 215Ah string out of 430Ah's in series/parallel) while partly depleted.
You are "testing" the system by randomly adding and removing loads, trying to determine which load is the "problem". In order to test, you need test equipment, plain and simple.
What was Vbatt at, when the alarm sounded? How many amps are actually getting to the batteries? What is the draw of each of the loads? My guess is you do not know... therefore, you too are guessing. So now everyone's guessing. It's the cabling. No, it's the freezer. No, it's the inverter. No, it's the laptop. No, it's the alternator... NO, IT'S A LACK OF INFORMATION! But good luck with all that.