If you do not have nothing between the 30A fuse and the 20A from the inverter ( like the pic ) the only thing that occurs to me is that you had a faulty 20A fuse on the battery side before.
But, I do have a very similar setup 40A fuse battery side ( like this
http://www.cartft.com/shop/catalog/il/19 ) and a 25A inline on the inverter side. The behaviour is the same. Beep but no reboot.
Latter I installed a 30A relay close to the inverter witch is used ( controlled by an itps ) as main power cut-off. Still works flawless ( if you exclude the problem I’m having with the inverter overheating and shutting off. It's kind of hot around here
)
I did measure the voltage at the inverter and I got 9.9v while cranking. Maybe you could test this by measuring at the battery fuse and at the inverter fuse while cranking to check if a 20A fuse can cause the voltage to drop.