AC300 grounding through house outlet?

Related to threads about grounding Bluetti connected devices…
I have my Bluetti AC300 plugged into grid at all times (PV Priority UPS mode) and power a few devices including a PC and furnace. I use a bonding plug as required by furnace.
Despite grid power in, I do NOT have continuity between my AC300 outlet grounds and my house ground (which makes sense otherwise a transfer switch would result in multiple neutral/ground bonds = BAD). This seems to indicate the grid charge ground is isolated from the Bluetti outputs.
Do I need aground for safety with metal cases? And if so, can use a ground-only suicide cord from my Bluetti to a house outlet to establish a ground (since I’m powering some devices that have a metal cases). See similar in this video - https://youtu.be/5WYxTq0AMkE?si=_PAdf4jrByHE4myK&t=468).
The BLUETTI outlet grounds all share continuity.
Thoughts - am I correct here?