If the self is scientifically measured, what is the Buddhist view on this?

Relational QM

I suggest that the common unease with taking quantum mechanics as a fundamental description of nature (the measurement problem) could derive from the use of an incorrect notion, as the unease with the Lorentz transformations before Einstein derived from the notion of observer-independent time. I suggest that this incorrect notion that generates the unease with quantum mechanics is the notion of observer-independent state of a system, or observer-independent values of physical quantities.

Dependent origination says exactly this, if only with different language, especially Nagarjuna's. Rovelli's QM is said to be (it is one of many interpretations, but the only which has no EPR paradox issue) a complete description of physical systems, same as dependent origination is a complete description of all conditioned phenomena. The Buddha in addition described the unconditioned element, which still eludes science entirely (as it must, as the scientific method is dualistic, requiring both observer and observed).

The theory of Self is a classical, even Newtonian concept. It is not 'wrong', simply less explicative and more prone to error than not-Self.

As for the direct question as to what the Buddha would say, this is a nice parallel to MN22:

“Bhikkhus, you may well acquire that possession that is permanent, everlasting, eternal, not subject to change, and that might endure as long as eternity. But do you see any such possession, bhikkhus?”—“No, venerable sir.”—“Good, bhikkhus. I too do not see any possession that is permanent, everlasting, eternal, not subject to change, and that might endure as long as eternity.

“Bhikkhus, you may well cling to that doctrine of self that would not arouse sorrow, lamentation, pain, grief, and despair in one who clings to it. But do you see any such doctrine of self, bhikkhus?”—“No, venerable sir.”—“Good, bhikkhus. I too do not see any doctrine of self that would not arouse sorrow, lamentation, pain, grief, and despair in one who clings to it.

“Bhikkhus, you may well take as a support that view that would not arouse sorrow, lamentation, pain, grief, and despair in one who takes it as a support. But do you see any such support of views, bhikkhus?”—“No, venerable sir.”—“Good, bhikkhus. I too do not see any support of views that would not arouse sorrow, lamentation, pain, grief, and despair in one who takes it as a support.

We are fortunate to live in a time where science had made impressive progress and has confirmed many aspects of the Buddha’s world view, but to think and believe that the only way to confirm what the Buddha taught is to see whether those teachings are compatible with science, is the wrong way to go.

There are many, many things that have not been “discovered” by science (or philosophy) yet, and the depth that the Buddha Dhamma had gone in understanding the MIND has not even been touched by science yet. Therefore, trying to gauge the validity of Buddha Dhamma using only the known facts from science is like a blind man trying to figure out what an elephant looks like by touching a leg of the elephant. Science can accept only those phenomena that can be observed and measured with scientific instruments. Such scientific instruments are basically “extensions” for our six senses.

Buddha Dhamma is self consistent. There is self-consistency within the foundational concepts, such as Tilakkhana, Four Noble Truths, Noble Eightfold Path, Paticca Samuppada, etc. Unlike Budda Dhamma, scientific explanations can change with time. Buddha has said that there is no “permanent” or “unchanging” self (which is called “soul” or “athma”) that remains the same throughout different births. Hence Buddha rejected both “permanent self” and “no self” concepts.

In current English translations, Sakkaya ditthi is translated as “self view” or “personality view” or “identity view”, etc. However all these translations have now come under scrutiny. As per the new interpretation given, Sakkaya = Sath + kaya (Sathkaya). “Sath” means – extremely valuable or useful or fruitful (the word “sath” has been derived from “atta” or “artha” which means meaningful or useful). And here the word “kaya” means the six pairs – the eye and the external sights, the ear and the external sounds, the nose and the external smells, the tongue and the external tastes, the body and the external contacts, the mind and the external dhammas. In other words here “kaya” means all the six internal rupa (sense faculties) together with all the external rupa. Hence Sakkaya ditthi is a wrong view that- all these “kaya” (all the internal as well as external rupa) are “sath” i.e., highly valuable or useful or fruitful or beneficial.

As stated in the OP, there is connotation of a “self”. Since we use names to label a person, that automatically gives the impression of a non-changing “self”. Thus it is a bit hard to remove this “sense of a self” from our minds. The Buddha said it is wrong to believe that there is a “self” and it is also wrong to believe that “there is no-self”. This is a bit difficult to comprehend first; that is why the Buddha said, “My Dhamma has never been known to the world…”.