= Solution
Multiply the supplied <Feynman gauge> propagators before reducing the numerator. Their two factors $-i$ and the source's <Clifford algebra> imply
$$
\gamma^\nu\widetilde S(p+l)\gamma^\mu\widetilde\Delta_{\mu\nu}(l)
=\frac{(d-2)(\not p+\not l)+dm}
{(l^2-i0)[(p+l)^2+m^2-i0]}.
$$
For example, contracting the numerator before the propagator factors gives $\gamma^\nu[-(\not p+\not l)+m]\gamma_\nu=-(d-2)(\not p+\not l)-dm$; the extra minus from $(-i)^2$ reverses both signs.
Use a <Feynman parameter> $x$ on the massive denominator and shift $q=l+xp$. The combined denominator is
$$
x[(p+l)^2+m^2]+(1-x)l^2=q^2+D(x),\qquad
D(x)=xm^2+x(1-x)p^2.
$$
The term odd in $q$ integrates to zero in <dimensional regularization>. The pole can be evaluated away from on-shell infrared singularities; its local ultraviolet coefficient is independent of $D$. With the source's master integral,
$$
\begin{aligned}
\mathcal I_{\rm UV}(p)
&=-\frac{ie^2}{8\pi^2\epsilon}\int_0^1dx\,[2(1-x)\not p+4m]\\
&=-\frac{ie^2}{8\pi^2\epsilon}(\not p+4m),\qquad
\Sigma_{\rm UV}(p)=\frac{e^2}{8\pi^2\epsilon}(\not p+4m).
\end{aligned}
$$
Terms proportional to $d-4$ multiply the pole into a finite term, so they do not change this result. The <counterterm> <kernel> from 1ai is $K_{\rm ct}=-\delta Z_2\not p-\delta m_L$. Cancelling $-\Sigma_{\rm UV}+K_{\rm ct}$ therefore gives
$$
\boxed{\delta Z_2=-\frac{e^2}{8\pi^2\epsilon},\qquad
\delta m_L=-\frac{e^2m}{2\pi^2\epsilon}.}
$$
The distinction between <mass> definitions matters. Since $Z_2m_0=m+\delta m_L$,
$$
\boxed{m_0-m=\delta m_L-m\delta Z_2=-\frac{3e^2m}{8\pi^2\epsilon},\qquad
Z_m-1=-\frac{3e^2}{8\pi^2\epsilon}\quad(m_0=Z_mm).}
$$
Equivalently the mass-dependent part of the <counterterm> density is $+e^2m\bar\psi\psi/(2\pi^2\epsilon)$, because its definition includes the explicit minus sign in $-\delta m_L\bar\psi\psi$. In the alternative convention $d=4-2\varepsilon$, replace every $1/\epsilon$ here by $1/(2\varepsilon)$. These are ultraviolet minimal-subtraction poles, not the finite on-shell <mass> correction. The <Ward identity> also fixes $\delta Z_1=\delta Z_2$; the <photon> coefficient $\delta Z_3$ requires its own two-point function and is not obtained from this graph.
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