Computational Complexity of Integer Linear Program [with Fixed number of 'Pure Constants']
Yes, it's still NP-hard.  Suppose the two allowed values for $c_0$ are 0 or 1, and that you're allowed to have equations of the form

$$c_0 + c_1 x_1 + \dots + c_k x_k = 0$$

as well (not just inequalities).  Suppose that we wish we could use the constant 17.  We can introduce new variables $z_1,\dots,z_6$, add the equalities below, and replace all occurrences of 17 with $z_6$.

$$\begin{align*}
1 - z_1 &amp;= 0\\
0 + 2 z_1 - z_2 &amp;=0\\
0 + 2 z_2 - z_3 &amp;=0\\
0 + 2 z_3 - z_4 &amp;=0\\
0 + 2 z_4 - z_5 &amp;=0\\
0 + z_1 + z_5 - z_6 &amp;=0
\end{align*}$$

Notice how the above equations force $z_1=1$, $z_2=2$, $z_3=4$, $z_4=8$, $z_5=16$, and $z_6=17$.

Of course there's nothing special about 17; you can do this for any other number as well.  In general, if you have any constant $\kappa$  you wish you could use, you can find an addition chain for $\kappa$, then use that to introduce temporary variables $z_1,\dots,z_t$ that force $z_t=\kappa$, then replace all occurrences of the constant $\kappa$ with the variable $z_t$.

In this way you can eliminate all constants other than 0 or 1 for $c_0$.  The length of the addition chain for $\kappa$ will be at most $2 \lceil \lg \kappa \rceil$, so the number of extra variables and equalities added is not too large (and in particular is polynomial in the size of the original problem), so this increases the size of the instance by at worst a polynomial factor.