Morning update:
The alarm continued to go off intermittently, for short time periods. After the first time I could not get to it before it shut itself off (the alarm, not the inverter) I realised I didn't have to.
This was when I had plugged in: phone charger, laptop charger, mattress pad, and heating pad (yes, both!). I unplugged phone charger, that did not help. I unplugged laptop charger and heating pad simultaneously, and that fixed it. So the inverter hates one of those two, and hates the freezer.
I can ascertain which one, of course, but my question is why? Sure, these low-power items have startup surges. But how big can those surges be? And what can I do about the freezer?