Two things. Most interactions reported would in scientific terms show up neurologically.
Not really, unless you are only focussing on humans/living things. There are plenty of interactions - e.g. gravity, light, spins etc that would result in interactions that aren't anything to do with neurology, although scientists, as smart people, might interpret those interactions neurologically. But those interactions would still occur regardless of any observation by humans or any other living thing.
One thing about the necessary entity though is that being non contingent means being a law unto itself ...
I don't think that is true at all. A necessary entity is one that is required to exist for other things to exist, in other words not contingent on another entity. That doesn't mean it is a law unto itself, still less that it somehow operates outside the rules that govern contingent entities. All it means is that contingent entities cannot exist/happen without it.
... so I still doubt how penetrable it is to science.
Of course it can - it may in itself be direct observable by science, or indirectly observed via its effects on contingent entities.
Which I believe relies somewhat on the repeatable.
Why wouldn't a necessary entity be consistent with repeatability - I would have thought that of all entities the most fundamental, the most required, the most necessary, will also be the most reproducible.
Secondly I wonder generally how the necessary entity fits into the laboratory or even field of study.
I do appreciate though your points.
Science constantly deals with identifying necessary and contingent elements in networks, pathways and systems (clearly on a more limited scale than universal) - so the standard approach is to remove elements and determine whether effects remain - if they do then the element you have remove isn't necessary, but may be contingent. If on the other hand removal means all further effects are abolished then this is necessary. Studies do this all the time, for example in medical research - knock out the action of a gene and look at downstream pathways.
The same approach can be applied more universally, albeit in a more theoretical manner as you can't easily knock-out an element of the big bang experimentally in the manner that you might with a gene. You can however, still conduct experimental studies, for example Hadron collider experiments aimed at identifying very short lived particle generation in conditions that may mimic those earliest event in the universe. Through these methods we can determine the inter-relationship between entities and elements and determine which may clearly be dependent on others and some which might be candidates for necessary entities.