question

As you haven't said what part of the world you are writing from it is hard to help as this a tricky area and subject to a lot of local variation.

However the earth terminals of the AC In, AC Out 1 and AC Out 2 are all connected together in the Multi (and to the chassis earth stud) so as a general rule it is probably best and simplest to have the earth conductors follow the respective live and neutral connections.

Bonding it all to the water pipes somewhere would for example be regarded in the UK as necessary (but not sufficient), and for island mode operation you would in addition need a separate ground spike, for a fuller treatment of the subject see here.

My instinct is since the Multi is neutral bond grounded at the Main Service panel, and the chasis ground is meant for grounding when the relay kicks in during inverting then the ground for the essential loads should be on the chasis?

The ground/PE is (should be) always connected.

The ground relay (when closed) connects the neutral to ground/PE.

I don't think so. Read sections 7.4 - 7.7 of Wiring Unlimited here. The figure on p60 shows the internal relays in the Multi. With ESS, if/when the grid fails the AC input relay L and N contacts open and the inverter is supplying the critical loads in island mode whether you like it or not.

A fraction of a second after this the neutral bond relay closes. This step is necessary because the critical load neutral has now lost the connection to the incoming panel where (as you say above) it was previously connected to earth. If this does not happen your protection against live-earth faults on these circuits is diminished or non-existent. IMHO you should not disable this facility, the provision is there for more unusual arrangements than we are discussing here. If your supplier disagrees please ask him to explain why. I do not understand "unlatching the Multi's ground relay will solve the issue" as I do not think you have an issue which this would solve.

You wrote <If I'm inverting to the essential loads panel does the AC OUT 1 ground float >. No, it should always be tied to system ground somehow, as I said before the easiest and most natural way of doing this (especially if you are using 3-core cable) is to take it to the AC OUT 1 earth terminal.

In the UK we are not allowed to use the incoming supply earth for operation in island mode (in case the supplier disconnects it during repair work), a separate earth spike is required as per the IET paper I linked previously (you should perhaps check locally to see if you need one where you are).

So in my own ESS system the Main Earthing Terminal in the incoming panel is bonded to the ground spike and also the water pipes. There is then a heavy link from the MET to the chassis stud in the Multi, which in turn provides the star point for all the inverter-fed circuits and the battery chassis. This arrangement is logical, hierarchical and avoids any ground loops.

Some confusion I think remaining between two different issues with the essential loads:

the ground bonding - which is a permanent hardwired arrangement (via the AC Out 1 terminal), you stil need to ascertain if you are required to have a separate earth electrode.

the neutral bonding - which is achieved in two ways, at the main service panel only when the grid is present, and by means of the neutral bond relay in the Multi when the grid is not present and the Multi is therefore disconnected from the main panel (unless you configure it without Loss of Mains detection which I strongly advise you to take local professional advice before doing, see the warnings here).

I have seen your other post about Assistants. It sounds extremely complicated and I don't think you will be able to do it with Assistants. You may be able to achieve some of what you want with Virtual Switch but probably you would need to control the Multi with Node Red or MQTT which is not entry-level DIY stuff. I would suggest you start by implementing ESS and getting to know what it can do. As I have found it is not perfect but you can set it up to achieve zero export and it is a good fit-and-forget solution for most circumstances.