Critical damping of SMPS LC filter
Actually Dr. Middlebrook shown that an optimum \$R_{damp}C_{damp}\$ combination exists to bring \$Q\$ down to the value you want and minimize instability risks. You can take a look at the maths behind the determination of the optimum \$RC\$ here.

I realized that I first thought you wanted to damp a front-end EMI filter but it seems that you want to take care of the buck output \$LC\$ filter output impedance. I honestly do not see the point because damping naturally occurs with feedback as long as the selected crossover frequency offers sufficient gain at the \$LC\$ filter resonant frequency.

Anyway, the thing is to determine the peaking you have without damping and what your target is when damped. In this quick example, I considered that my design target was an arbitrarily-selected 10-\$\Omega\$ resistance for the maximum peaking once damping elements are installed. However, it is very likely that the final peaking is less than this value considering all dissipation factors in the filter - \$r_L\$, \$r_C\$, magnetic losses and so on which will naturally lower the \$Q\$ giving more margin at the end. The calculation of the damping elements gives \$C_{damp}=36\mu F\$ and \$R_{damp}=6\;\Omega\$. You can appreciate how far this is from the 300-\$\mu F\$ cap. you first computed.

The below Mathcad sheet shows you how the computation is done and how the resulting output impedance is damped for this 3rd-order filter (2nd-order plus another damping cap. is 3rd order).





In the above graph, you see the resulting output impedance plot whose peak is constrained to 20 dB\$\Omega\$, exactly our design target.

So in your case, you first need to determine the peak output impedance without damping and how low you want to lower the \$Q\$ once damping elements have been installed. Calculations are not trivial considering the 3rd-order filter, but the PPT given in the link contains all the elements you need.