Even more is true: Two possible cores of $M$ lie in the same genus, i.e. they become isomorphic in the localization at any prime of $\mathbb Z$. However, I couldn't find an explicit proof of this anywhere in literature Edit: It's Proposition 5.4 in Gruenberg: "Relation Modules of Finite Groups" (though unfortunately there's apparently no online version of this book besides google books).
Assume you have two decompositions $M=C_1\oplus P_1 = C_2 \oplus P_2$ with $P_1$ and $P_2$ projective. Assume furthermore that one cannot split off any further non-zero projective summands from $C_1$ and $C_2$.
By a theorem of Swan (see Swan: Projective modules over finite groups) $\mathbb Q\otimes P_1$ and $\mathbb Q\otimes P_2$ are free, and therefore so is $\mathbb Z_{(p)} \otimes P_1$ and $\mathbb Z_{(p)}\otimes P_2$ ($\mathbb Z_{(p)}$ denotes the localization of $\mathbb Z$ at $p$; this simply uses the fact that since the columns of the decomposition matrix of a finite group are linearly independent, the isomorphism type of the $\mathbb Q$-span of a projective $\mathbb Z_{(p)}$-module determines the isomorphism type of the projective module; alternatively see Theorem (77.14) on p. 548 in Curtis, Reiner: Representation theory of finite groups and associative algebras).
So now apply cancellation for $\mathbb Z_{(p)}G$-lattices (see Corollary 1 in this paper of Reiner) to find that unless $P_1$ and $P_2$ have the same $\mathbb Z$-rank (in which case we would be done), either $C_1$ or $C_2$ (say, w.l.o.g. $C_1$) has a non-zero free summand in every localization. So if this is the case, then take for each prime $p_1,\ldots,p_n$ that divides $G$ an epimorphism
$$
\phi_i: \mathbb Z_{(p_i)} \otimes C_1 \twoheadrightarrow \mathbb Z_{(p_i)}G \subset \mathbb QG
$$
By multiplying each $\phi_i$ with a sufficiently large integer relatively prime to $p_i$ we may assume $\phi_i(C_1) \subseteq \mathbb Z G $ for each $i$. Now we get a map
$$
\phi := \sum_{i=1}^n \left(\prod_{j\neq i} p_j \right)\cdot \phi_i|_{C_1}
$$
$\phi$ is a homomorphism from $C_1$ to $\mathbb Z G$, which is an epimorphism in the localization at each $p_i$ (by construction: the factor $\prod_{j\neq i} p_j$ is a unit in $\mathbb Z_{(p_i)}$ and in the radical of each $\mathbb Z_{(p_j)}$. Hence the image of $\phi$ in the localization at $p_i$ is the same as the image of $\phi_i$). The image of $\phi$ in the localization at any other prime is projective anyways, since if $p$ doesn't divide the order of $G$, then any $\mathbb Z_{(p)}G$-lattice is projective. So now we have a homomorphism from $C_1$ onto the image of $\phi$, and the image of $\phi$ is projective in each localization. But then by Theorem (78.1) on p. 550 in Curtis&Reiner the image of $\phi$ is projective, and therefore the image of $\phi$ is a projective summand of $C_1$. This contradicts the assumption that we couldn't split off any further projective summands from $C_1$.
