Is there a metric on $\mathbb{R}^2$ such that the unit circle is a geodesic?

There are many possibilities.

For example, you can use a stereographic projection to map $\mathbb R^2$ to a sphere of unit diameter minus a point, such that the unit circle maps to the equator. Then pull the metric on the sphere back to $\mathbb R^2$. This gives a nice conformal metric: $$ ds^2 = \frac{dx^2+dy^2}{(x^2+y^2+1)^2} $$

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Intuitively I would expect that whenever you have $ds^2=f(x^2+y^2)^2\cdot(dx^2+dy^2)$ for some function $f:\mathbb R_{\ge 0}\to\mathbb R_{>0}$ that falls off to $0$ "sufficiently fast", there will be some circle around the origin that is a geodesic. With some luck, this might allow you to choose an $f$ of a form that makes your subsequent computations simpler.