I work out the case $d=2$ below. I didn't check everything carefully, so hopefully there are no errors.

Up to homothety, any lattice is equivalent to one generated by the complex numbers $1,z$ with $z \in \mathbb{H}$. In fact, $z$ can be chosen to lie in the standard fundamental domain for $SL_2(\mathbb{Z}) \backslash \mathbb{H}$. To make such a lattice unimodular, we simply re-scale by a scalar $\lambda > 0$ to get $\lambda, \lambda z$ with $\lambda = y^{-1/2}$. Here $z= x +i y$, $y>0$. Then any element of the lattice $\Lambda$ may be written uniquely as $\lambda cz + \lambda d $ with $(c,d) \in \mathbb{Z}^2$, $(c,d) \neq (0,0)$. The sum in question, in this notation, is then
$$\sum_{0 < |\lambda cz + \lambda d | < R} |\lambda cz + \lambda d |^{-2}.$$
By a Perron-type formula, we can evaluate such a sum asymptotically by a contour integral of the form
$$\lim_{T \rightarrow \infty} \frac{1}{2 \pi i} \int_{\sigma - iT}^{\sigma + iT} R^s F(s) \frac{ds}{s},$$
where $$F(s) = \sum_{(c,d) \neq (0,0)} |\lambda cz + \lambda d |^{-2-s}.$$
In practice, all that matters is the analytic behavior of $F(s)$ near $s=0$.

Now $F(s)$ is closely related to the Eisenstein series $E(z,s)$ defined by
$$E(z,s) = \frac{1}{2} \sum_{\gcd(c,d) =1 } \frac{y^s}{|cz+d|^{2s}} = \frac12 \frac{1}{\zeta(2s)} \sum_{(c,d) \neq (0,0)} \frac{y^s}{|cz+d|^{2s}}.$$
Unless I made a mistake, a short calculation (pulling out a gcd to give the zeta function) gives $F(s) = 2 \zeta(2+s) E(z,1+\frac{s}{2}).$

The constant $c_1$ only depends on the residue of $F(s)$ at $s=0$, which one can surely calculate quite easily; it does not depend on the lattice of course. To get $c_2$ one needs to calculate the next term in the Laurent expansion of $F(s)$, which I believe equals a constant minus $\log y^{1/2} |\eta(z)|^2$. Here this function $f(z)=\log y^{1/2} |\eta(z)|^2$ is $SL_2(\mathbb{Z})$-invariant. Now $f(z)$ depends on $z$, and so yes $c_2$ depends on the lattice in a rather interesting way.

I wager that for $d \geq 3$ one needs to find the relevant Eisenstein series and its Laurent expansion.