If you look into the original graphics you may observe that the shaded area is only from the middle to the left and from the middle to the right it is almost completely black. So dividing the graphics in two and building on the suggestions by Peter Grill, the SVG paths defined by the OP, and others, here is one solution that moves the entire curve to the right:
\documentclass{minimal}
\usepackage{tikz}
\usetikzlibrary{svg.path}
\usepackage{spath} % Package for manipulating paths by Andrew Stacey
% The curve that will be repeated to the left at a lower-tone.
\newcommand{\MiddleC}{%
[xshift=25] svg {M201.1,673.2c1.4-39.8,2-52.2,18.2-70.8c11.7-13.5,18.3-24.3-7.7-49.4
c-39.3-38-43.2-59.2,1-90.2c18.1-12.7,67.1-20.2,25.4-36.6c-18.2-7.2-23.5-9.7-26.9-39.2c-3-26.7-17.7-28.2-37.7-35
c-20-6.9-87.7-28.8-50.2-78.2c23.5-31,58.6-73.9,83.1-118.2c13.3-24,22.4-56.5,38.6-85.9};
}
% The repetition of this curve, by Peter Grill - answer 1.
\newcommand{\MyScope}[1][]{%
\begin{scope}[cap=round, #1]
\path[draw=black] \MiddleC;
\end{scope}
}
\newlength{\shift}
\begin{document}
\begin{tikzpicture}[transform canvas={yscale=-1}] %SVG curves were upside down.
\path [save path=\Left] svg {M201.1,673.2c1.4-39.8,2-52.2,18.2-70.8c11.7-13.5,18.3-24.3-7.7-49.4
c-39.3-38-43.2-59.2,1-90.2c18.1-12.7,67.1-20.2,25.4-36.6c-18.2-7.2-23.5-9.7-26.9-39.2c-3-26.7-17.7-28.2-37.7-35
c-20-6.9-87.7-28.8-50.2-78.2c23.5-31,58.6-73.9,83.1-118.2c13.3-24,22.4-56.5,38.6-85.9};
\path [save path=\Middle, xshift=25] svg {M201.1,673.2c1.4-39.8,2-52.2,18.2-70.8c11.7-13.5,18.3-24.3-7.7-49.4
c-39.3-38-43.2-59.2,1-90.2c18.1-12.7,67.1-20.2,25.4-36.6c-18.2-7.2-23.5-9.7-26.9-39.2c-3-26.7-17.7-28.2-37.7-35
c-20-6.9-87.7-28.8-50.2-78.2c23.5-31,58.6-73.9,83.1-118.2c13.3-24,22.4-56.5,38.6-85.9};
\path [save path=\Right] svg {M295,69.4c-5.1,25.7-13.3,57.3-19.2,76.5
c-15,48.3-21.9,52.2-36.7,86.5c-21.5,49.9-47.2,77.8-1.1,90.5c46.7,12.8,49.7,16,48.9,40.6c-0.6,18.4-3.9,35.1,10.3,34.3
c15.7-0.9,40.5,6,71.5,36.8c8.2,8.1,25,18.7,25,18.7s-159.2,8.6-116.4,73.2c29.6,44.7,31.9,44.4,19.9,66.7
c-15.1,28.3-28.8,48.8-23,80.6};
% Concatenating the 4 paths so we can get the black part:
\pgfoonew \patha=new spath(\Middle)
\pgfoonew \pathb=new spath(\Right)
\patha.concatenate with lineto(,\pathb)
\patha.close()
% Drawing and filling the black part:
\patha.use path with tikz(line width=3pt,draw=black,fill=black)
% Repeating the path for the shaded part:
\foreach \x in {1,...,200} {
\pgfmathsetlength{\shift}{\x/2 pt}%
\pgfmathsetmacro{\opacity}{1-\x/100}%
\MyScope[xshift=-\shift, line width=0.5pt, draw opacity=\opacity]
};
% The eye of the bird:
\filldraw[left color=white, right color=white, draw=white] (11.9,15.1) circle [radius=7pt];
\end{tikzpicture}
\end{document}
which yields:
Observe that applying with a gradient shade in between two curves would, in general, require a path from each point of one curve to a point on the other, which can be quite cumbersome if one of them had self-intersections or a horizontal line would intersect a curve in two or more points.
In the case at hand, horizontal lines intersected the curves in one point only each, making the job a bit easier.
The proposed solution of using a "buffer" zone in only one color would allow one to deal with more complex cases till the curve could be made more straight and then slide it to the other.
\documentclassand the appropriate packages that sets up the problem. In this specific case, an MWE that draws the path that you want the gradient applied to would be sufficient. While solving problems can be fun, setting them up is not. Then, those trying to help can simply cut and paste your MWE and get started on solving the problem.