Method for finding the best non-increasing approximation function of a given signal
You will find some pointers with the correct wording. There is a lot of statistical literature on monotone or monotonic regression (sometimes called isotonic regression). A more generic term is "shape-constrained estimation".

For instance, a few references:


Constrained statistical inference: inequality, order, and shape restrictions, 2005, Mervyn J. Silvapulle and Pranab Kumar Sen
Polynomial algorithms for isotonic regression, 1997, Victor Chepoi and Daniel Cogneau and Bernard Fichet
Fitting monotonic polynomials to data, 1994, Douglas M. Hawkins


The most common version is a least square fit with piecewise constant or linear functions, see for instance Scikit-learn: Isotonic Regression




  An illustration of the isotonic regression on generated data. The
  isotonic regression finds a non-decreasing approximation of a function
  while minimizing the mean squared error on the training data. The
  benefit of such a model is that it does not assume any form for the
  target function such as linearity. For comparison a linear regression
  is also presented.


With more recent progress in optimization, there are works on fitting under other norms ($\ell_1$, $\ell_\infty$) and contraints on the lower or upper bound of the derivative, which should be either positive or negative, to impose non-increasing or non-decreasing approximations.

[EDIT:2018/12/24] Now, on the question whether the signal should be low-pass or not: if you think that a paramteric function is  too restrictive, a low-pass filter would probably too restrictive as well. Indeed, how do you know that the characteristics or the filter is sufficient? wgy should it be linear? does it cope with the nature of the signal and noise well enough?