Why is it necessary at all to define physical processes “exactly”? — They are anyway superficial interactions, are circularly defined (not to speak yet of their measurement) and may hence not have foundations in itself.
That the physical relations (or reality) in their interactions and the measurements thereof are mutually circularly defined, is not severe. The problem is rather that both the interacting physical objects and the physical measurements (which are, as you said rightly, of the same essence) are superficial relations.
This is not the case with chemical processes, which are deep acting genuine relations (which may be presented in bottles); hence physical interactions could anyway not be the foundations of physics. By the way they are anyway circular in itself (not to speak yet of measurements or feelings), as has been described in this post (the genuine relations of chemistry are explained in the link there quite below).
The mere facts of
1. the circularity of the mathematical relations (of interacting physical objects) i.e. of physical concepts itself, and
2. the circular dependence of this physical reality on the measurement of them,
are not a problem. It is the mentioned superficiality, which leads to the problem that physics may only be described by numbers or number-variables, and that we have hence no foundations in physics. It is not circularity of any kind (whether by maths itself, or in relation to measurements or feelings), which is the real problem. To draw an analogy: The problem is circularity without an Archimedian point.
— And now you may vent your anger by down voting me because you will not grasp what I say since philosophers, scientists, politicians are themselves grasped by their projections, which I have already hinted in other posts here.