 Pinned Brownian loop measure at an interior point

In the unit disc, the relevant pointed Brownian loop measure is rooted at $0$ and normalized by its conformal deficit: for simply connected $U$ containing $0$, its mass of loops not contained in $U$ is minus the logarithm of the conformal radius. Its outer-boundary pushforward gives the corresponding pointed simple-loop restriction measure. It is not the full unrooted loop measure or a probability law of a fixed-duration Brownian bridge.