This is basically an extended comment:
First, it is important to note that techniques from QFT have been applied to the problem of hydrodynamical turbulence for some time. Einstein's last postdoc, Robert Kraichan, used them in formulating his DIA (Direct Interaction Approximation) formalism - likely still the greatest theoretical insight into turbulence besides Kolmogrov's 1941 work and the early work of Reynolds/Taylor/Prandtl. I would highly recommend the book "A Voyage Through Turbulence" to get a grip on some of the history and techniques used in this quagmire of a subject (there are lectures associated with this book on youtube as well, that may be found here).
However, QFT is inherently different than hydrodynamical turbulence because there is scale separation (the particle interactions in QFT are taken to be $weak$), hence one can truncate the order of interaction and converge to the observed physics. That is, let the Lagrangian be given by (in the sense of Hamilton's principle)
where $\epsilon$ is some small parameter (that is a dimensionless ratio of physical quantities arising in the equations of motion - for example the ratio of a mean flow to a perturbation). In QFT, $\epsilon<<1$ and one may truncate $L$ after a certain number of terms (say, 4) and get a mathematical theory that converges towards the physical reality that is observed.
The same thing, dressed slightly differently, occurs in turbulence, where a $closure$ scheme is needed to solve the moment equations. The methods typically used, eg quasinormal assumptions, are analogous to making assumptions on strength of interactions. However, in turbulence these interactions are not weak (and a series like the one written above for $L$ may not converge), and hence these approximations do not fully describe observations.
Getting to your question - using a variational approach won't simplify any of the issues mentioned above, as at some point approximations (even if they are at the order of the action) must be made. However, from the perspective of numerical analysis, working with a Langrangian or Hamiltonian can be very useful in fluid mechanics as it allows you to know a priori what the conservation laws are of the truncated system, serving as additional tests on the validity of the model output.
The paper you referenced is idiosyncratic and does not present a Lagrangian for the full equations of motion (he makes an ansatz about the pressure term). The best review on Hamiltonian Fluid Mechanics is given by Salmon (1988).
There are examples of turbulence which are effectively described by field theoretical methods using a Hamiltonian because they possess scale separation. In particular, weak (where the waves are assumed to have small slope and be narrow-banded) wave turbulence of surface waves is examined in this fashion (Zakharov won a Dirac medal for this in 2003, along with Kraichnan).