I think you are going in the right direction. But let me push even a bit further.
Everyone takes as a given that they know themselves to be sentient. but let me push back on this: Consider how odd it is that many have strong conviction that they are this thing: Aware/Conscious, yet often they cannot even define these terms to their own satisfaction. Its like a Dr. Seuss book:
I am most definitely a kerbloofer! and what is that, you ask? Well I am not sure, except to know that I am one!
What is going on here? Really this term 'awareness' is a placeholder for an effect perceived, which the perceiver is at a loss to really pin down, except for a confidence of what the effect is NOT like. Specifically the perceiver knows that this effect is quite distinct from any other forms of information or experience that it has sensed.
This all we really 'know' about our own sentience.
Just as we cannot know what anyone elses perception of a red wall is, we cannot know what their perception of their own awareness is.
but we can do an analysis of other biological and non-biological systems to see if they also have draw the same distinctions about the world they sense.
And if such systems also have a distinct sensation we can further query the systems beliefs about this distinct sensation.
So we actually can potentially come to know that another entity is sentient with similar confidence that we have about ourselves (e.g. the sense a distinctive pattern about which they derive similar conclusions that we derive about our own pattern.)
I am removing the subjectivity here regarding what is measured. The thing objectively measured in all cases is the presence of a distinct category of sensing having a collection of the right properties.
for example this distinctive sensing seem causally antecedent to other fact knowlege that we know.
Maybe such systems ARE "zombies" but then maybe I am too. In both cases all we can know is that the entity itself has concluded that it is not zombie... e.g. it has special perception.
I think if we run down the list of things we want full human consciousness to have, LamDA will fall short, still the next system need not fail.
