Connecting homomorphism and fibre transfer

Question

I have a very special situation: Consider an orientable surface bundle $F_g\to E\stackrel{\pi}{\to} B$ with fibre a closed orientable surface of genus $g$. We have the “integration along the fibre” $$\pi^!:H^2(E)\to H^0(B)$$ Consider a section $s:B\to E$ and the open subbundles $U:=E\setminus s(B)$ and $V$ a tubular neighborhood around $s(x)\in E_x$ in each fibre. Then $(E;U,V)$ is an excisive triad and $U\cap V\simeq \partial V$. Consider the Mayer–Vietoris connecting homomorphism $$\delta^*:H^1(U\cap V)\to H^2(E).$$ Now let's assume that $\partial V\cong B\times \mathbb{S}^1$ is the trivial $\mathbb{S}^1$-bundle over $B$. Denote $\varrho:\partial V\to B$. We can consider the “integration along the fibre” here, $$\varrho^!:H^1(U\cap V)\cong H^1(\partial V)\to H^0(B).$$ I conjecture that these maps fit together, which means $$\pi^!\circ\delta^* = \varrho^!$$ However, I don't know how to show this, because I don't know how to relate the two transfer maps (which I only defined as the edge morphisms in the corresponding spectral sequence).