If I understand your definition of area correctly, the Brownian bridge ends up having infinite filling area because of small regions with large winding number -- a result of Yor shows that the expected area of the region with winding number $k$ is $\sim 1/k^2$. This contributes $\sim 1/k$ to the area of a filling, so the expected area is infinite.

On the other hand, there are asymptotic bounds on the growth of the area of the random walk bridge. I wrote a paper with Bruno Schapira which shows that the expected homological filling area of a random walk bridge of length $n$ grows like $n \log \log n$. (The homological filling area of a curve is the area of the smallest surface of arbitrary genus whose boundary is the curve -- in this case, it's the integral of the absolute value of the winding number over the plane.)

Roughly, we estimated the homological filling area of the random walk bridge by estimating the area of the region with winding number $k$ for all $k$. A rescaled random walk stays close enough to a Brownian motion that the region with winding number $>k$ has expected area $\sim 1/k$ when $k\le \log n$. When $k>\log n$, the area drops off quickly, so the homological filling area grows like $\log \log n$.