Let us first introduce the notion of a quasi-sheaf with compact support.

Def. Let $\mathcal F$ be a presheaf on a locally compact space $X$. We call it a quasi-sheaf with compact support if the following holds:

(A1) For every open $U\subset X$ and every section $s\in \mathcal F(U)$, $\overline{\mathrm{supp}(s)}$ is compact.

(A2) Let $U_i,i\in I$ be a family of opens, $s_i\in\mathcal F(U_i)$, for every two $i,j$ the restrictions of $s_i$ and $s_j$ to $U_i\cap U_j$ coincide and $\overline{\bigcup\limits_{i\in I}\mathrm{supp}(s_i)}$ is compact. Then there exist a section $s\in \mathcal F(\bigcup\limits_{i\in I}U_i)$ such that its restrictions on $U_i$ are $s_i$ and this section is unique.

Let us denote the category of quasi-sheaves with compact support on a locally compact $X$ by $QSh_c(X)$. Then the forgetting functor $QSh_c(X)\to PreSh$ has a left adjoint which we call a quasi-sheafication with compact support and denote by $\mathcal F^c$

Claim. $QSh_c(X)$ is an abelian category.

Let $p:X\to Y$ be a continuous map. Then it induces functors $p_*\colon Qsh_c(X)\to Qsh_c(Y)$ and $p^*\colon QSh_c(Y)\to QSh_c(X)$ if $p$ is proper..

Now we move to cohomology of qswcs.

Claim. $QSh_c(X)$ has enough injectives.

Def. (1) A qswcs is called flasque if $\forall U$ the map $\Gamma(\mathcal F)\to \mathcal F(U)$ is surjective.

(2) A qswcs is called soft if for every compact subset $K$ the natural map $\Gamma(\mathcal F)\to\Gamma(i_*i^*\mathcal F)$ where $i$ is the inclusion of $K$ in $X$ is surjective.

Claim. (1) Injective qswcs are flasque.

(2) Flasque qswcs on a paracompact space are soft.

(3) Flasque and soft qswcs on a paracompact space are acyclic.

Thm. Let $\underline{\mathbb R}$ denote the constant qswcs on a manifold $M$. Then $H^*(\underline{\mathbb R}) = H^*_c(M)$.

Now we are in position to compute the zeroth and first layer of Serre spectral sequnce. We can try to mimic the proof given in Bott, Tu for qswcs. The zeroth sheet of a spectral sequence is

$$E_0^{p,q} = \Gamma(\mathcal C^p(p_*\Omega^q(E)^c),$$

where $\mathcal C^q(\mathcal F)$ is the $q$-th term in the Čech resolvent of $\mathcal F$ associated with a cover of $M$ by small enough contractible compact subsets.

It converges to cohomology with compact support of $E$.

Now we compute the first sheet of the spectral sequence modulo two conjectures.

Firstly, we use the following conjecture which I can't prove formally but I think it follows from abstract homological algebra.

For each exact sequence of acyclic qswcs $\mathcal G^*$ we can form a qswcs in the following way. We have a family of presheaves
$$
U \mapsto \frac{ker\: d\colon \mathcal G^n(U)\to \mathcal G^{n+1}(U)}{im\: d\colon \mathcal G^{n-1}(U)\to \mathcal G^n(U)}
$$

Now take a quasi-sheafication with compact support of this presheaf and denote it by $\mathcal H^n(G^*)$.

Conj. (1) $H^n(\mathcal G^*):=\frac{ker\: d: \Gamma(\mathcal G^n)\to \Gamma(\mathcal G^{n+1})}{im\: d: \Gamma(\mathcal G^n)\to \Gamma(\mathcal G^{n+1})} = \Gamma(\mathcal H^n(\mathcal G^*))$

(2) $E_1^{p,q} = \Gamma(\mathcal C^p(\mathcal H^q(p_*\Omega^*(E))^c)$, i.e. taking cohomology commutes with forming the Čech complex.

Conj. Let $f,g\colon M\to N$ be two proper maps such that there exists a proper homotopy $H\colon M\times I\to N$ between them. Then they induce the same map on cohomology with compact support.

It follows that $\mathcal H^q(p_*\Omega^*(E)) = \underline{H_c^q(F)}$, hence

$$
E_1^{p,q}=\Gamma(\mathcal C^p(\underline{H_c^q(F)})^c)
$$