= Solution
The interaction contains no time derivatives, so $\pi=\dot\phi$. The <Legendre transform in mechanics> gives
$$
\mathcal H=\pi\dot\phi-\mathcal L
=\frac12\pi^2+\frac12(\nabla\phi)^2+\frac12m^2\phi^2+\frac{\lambda}{4!}\phi^4.
$$
Thus
$$
\boxed{\mathcal H_0=\tfrac12\pi^2+\tfrac12(\nabla\phi)^2+\tfrac12m^2\phi^2,
\qquad\mathcal H_I=\frac{\lambda}{4!}\phi^4,\qquad H=H_0+H_I.}
$$
Here $H_0=\int\mathcal H_0$ is the free <Hamiltonian operator> of mass $m$, and $H_I=\int\mathcal H_I$ is the <interaction Hamiltonian>.
In the <interaction picture>, operators evolve with $H_0$ and the states with $H_I(t)=e^{iH_0t}H_Ie^{-iH_0t}$. With $U_I(t_0,t_0)=I$, the state evolution satisfies $i\partial_tU_I(t,t_0)=H_I(t)U_I(t,t_0)$. Integrating once and iterating gives the <Dyson series>
$$
U_I(t,t_0)=I+\sum_{r\ge1}(-i)^r\int_{t_0<t_r<\cdots<t_1<t}dt_1\cdots dt_r\,
H_I(t_1)\cdots H_I(t_r).
$$
Equivalently it is the <time ordering> of the exponential. Taking the infinite-time scattering limit, with adiabatic switching understood, gives the <scattering matrix>
$$
\boxed{S=T\exp\!\left[-i\int d^4x\,\mathcal H_I(\phi_I(x))\right].}
$$
The field in the interaction is the free interaction-picture field, which is essential for applying the free-field contractions.
Use <relativistic normalization of a one-particle state>, $|\mathbf p\rangle=\sqrt{2E_{\mathbf p}}a^\dagger(\mathbf p)|0\rangle$, so each external scalar contraction contributes its plane-wave phase without an extra $1/\sqrt{2E}$ factor. For the connected on-shell two-to-two <scattering amplitude>, the first <Dyson series> term is $-i\lambda\int d^4x\,\phi_I(x)^4/4!$. The <Wick theorem> gives $4!$ ways to attach the two incoming and two outgoing external particles to the four fields, canceling the denominator. Consequently
$$
\langle p_3,p_4|(S-I)|p_1,p_2\rangle_{\mathrm{connected}}
=-i\lambda\int d^4x\,e^{i(p_3+p_4-p_1-p_2)\cdot x}+O(\lambda^2)
=i(2\pi)^4\delta^4(p_3+p_4-p_1-p_2)\mathcal A,
$$
where
$$
\boxed{\mathcal A=-\lambda+O(\lambda^2).}
$$
The delta function expresses <four-momentum conservation>. The connected scattering convention removes spectator, vacuum and external self-energy contributions; equivalently at this order one can normal-order the interaction and keep the physical external mass fixed. These contributions should not be mistaken for additional connected four-point diagrams.
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