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If you just give the mass an explicit time-dependence,
$$L = \frac12 m(t) \dot{x}^2 - U(x)$$
then the Euler-Lagrange equation is
$$\frac{d}{dt} (m \dot{x}) = \dot{m} \dot{x} + \dot{m} \ddot{x} = - \frac{dU}{dx}.$$
It's unclear to me why you think "we simply cannot use" this result. It isn't Galilean invariant, but once you let $m(t)$ have arbitrary time-...
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I’m answering the question you asked at the end:
Assuming physics solves its equations in symmetric way, e.g. QM with Feynman path integrals instead of Schrödinger equation, do Bell's assumptions hold - are local realistic "hidden variables" still disproven?
The short answer is “Yes.” Solving QM with path integrals is equivalent to using Schrödinger’s ...
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