I suspect that the answer ultimately comes from the behaviour of Chern classes under pullbacks. More precisely, if $L\rightarrow M$ is an $S^1$-equivariant complex line bundle with equivariant first Chern class $c_1^{S^1}(L)\in H_{S^1}^2(M)$, then the $S^1$-equivariant first Chern class of the restricted bundle $L\vert_{\mu^{-1}(0)}\rightarrow\mu^{-1}(0)$ is precisely $i^*(c_1^{S^1}(L))\in H_{S^1}^*(\mu^{-1}(0))$, where $i:\mu^{-1}(0)\rightarrow M$ is the inclusion. The image of $c_1(L)$ under the Kirwan map is then the image of $i^*(c_1^{S^1}(L))$ under the isomorphism $H_{S^1}^*(\mu^{-1}(0))\rightarrow H^*(\frac{\mu^{-1}(0)}{S^1})$.

I am not offering an answer, but I suspect that the answer ultimately comes from the behaviour of Chern classes under pullbacks. More precisely, if $L\rightarrow M$ is an $S^1$-equivariant complex line bundle with equivariant first Chern class $c_1^{S^1}(L)\in H_{S^1}^2(M)$, then the $S^1$-equivariant first Chern class of the restricted bundle $L\vert_{\mu^{-1}(0)}\rightarrow\mu^{-1}(0)$ is precisely $i^*(c_1^{S^1}(L))\in H_{S^1}^*(\mu^{-1}(0))$, where $i:\mu^{-1}(0)\rightarrow M$ is the inclusion. The image of $c_1^{S^1}(L)$ under the Kirwan map is then the image of $i^*(c_1^{S^1}(L))$ under the isomorphism $H_{S^1}^*(\mu^{-1}(0))\rightarrow H^*(\frac{\mu^{-1}(0)}{S^1})$.

I suspect that the answer ultimately comes from the behaviour of Chern classes under pullbacks. More precisely, if $L\rightarrow M$ is an $S^1$-equivariant complex line bundle with equivariant first Chern class $c_1^{S^1}(L)\in H_{S^1}^2(M)$, then the $S^1$-equivariant first Chern class of the restricted bundle $L\vert_{\mu^{-1}(0)}\rightarrow\mu^{-1}(0)$ is precisely $i^*(c_1(L))\in H_{S^1}^*(\mu^{-1}(0))$, where $i:\mu^{-1}(0)\rightarrow M$ is the inclusion. The image of $c_1(L)$ under the Kirwan map is then the image of $i^*(c_1(L))$ under the isomorphism $H_{S^1}^*(\mu^{-1}(0))\rightarrow H^*(\frac{\mu^{-1}(0)}{S^1})$.

I suspect that the answer ultimately comes from the behaviour of Chern classes under pullbacks. More precisely, if $L\rightarrow M$ is an $S^1$-equivariant complex line bundle with equivariant first Chern class $c_1^{S^1}(L)\in H_{S^1}^2(M)$, then the $S^1$-equivariant first Chern class of the restricted bundle $L\vert_{\mu^{-1}(0)}\rightarrow\mu^{-1}(0)$ is precisely $i^*(c_1^{S^1}(L))\in H_{S^1}^*(\mu^{-1}(0))$, where $i:\mu^{-1}(0)\rightarrow M$ is the inclusion. The image of $c_1(L)$ under the Kirwan map is then the image of $i^*(c_1^{S^1}(L))$ under the isomorphism $H_{S^1}^*(\mu^{-1}(0))\rightarrow H^*(\frac{\mu^{-1}(0)}{S^1})$.