[EDIT] The discussion below gets beyond the specific question raised here. I've looked further into the standard textbook literature, which is fairly limited in terms of giving detailed accounts of Brauer theory. In Serre's lectures (translated into English as Springer GTM 42), Exercise 15.2 is close to Ben's answer here, but similarly doesn't start with a *simple* module in characteristic 0. But the treatise by Curtis-Reiner *Methods of Representation Theory* in two volumes does have an exercise early in Chapter 2 (which introduces modular representations of finite groups in a modern style). This occurs on p. 416 at the end of $\S16$ (before the introduction of Brauer characters in $\S17$), as Exercise 3: "Find an example of a $p$-modular system $(K,R,k)$, a finite group $G$, and two full $RG$-lattices $M_1$ and $M_2$ in a simple $KG$-module $V$, such that $\overline{M}_1$ and $\overline{M}_2$ are *not* $kG$-isomorphic."
(Here the bar indicates reduction modulo $p$ and the notation differs from that used by spin.)
Probably they intended here a fairly small group such as $S_3$, known to have simple $KG$-modules of dimension $1,1,2$. For $p=3$ $kG$ has only the reductions of those of dimension 1 (trivial and sign) as simple modules, whereas the reduction modulo $3$ of the natural 2-dimensional module has these two as composition factors but apparently has the desired two contrasting choices of $RG$-lattice. Anyway, the authors viewed Exercise 3 as a fairly elementary exercise.
[It's unclear to me in general whether one might "usually" expect to get *indecomposable* and *completely reducible* (not just *decomposable*) $kG$-modules by varying the choice of $RG$-lattices in a simple $KG$-module.]
----------------------------------------------------------------------
To expand my brief comment, I can point to specific examples involving finite groups of Lie type in the defining characteristic $p>0$. The general set-up for reduction mod $p$ is somewhat complicated and involves a $p$-modular system in the Curtis-Reiner sense (a suitable valuation ring $R$ such as a ring of $p$-adic integers, with fraction field of characteristic 0 and finite residue field of characteristic $p$). But an example such as $G = \mathrm{SL}_2(\mathbb{F}_p)$ suffices. Here the irreducible representations and their characters over $\mathbb{C}$ have been known since the time of Frobenius; an elementary exposition is given <a href="https://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/Humphreys.pdf">*here*</a>. Take $p$ "large enough" to avoid minor problems.
For this finite group $G$, approximately half (= reciprocal of order of Weyl group) of the irreducibles belong to the "principal series" (induced from 1-dimensional characters of a Borel subgroup). In turn, reduction mod $p$ of such a representation typically (though not always) involves two composition factors of different dimensions. Here you get a concrete instance of the phenomenon you ask about. A natural choice of $RG$-lattice for a typical principal series representation leads to an indecomposable module over the finite field (here $\mathbb{F}_p$) having two composition factors: these correspond to adjacent edges in the Brauer tree of the block in question. But these irreducible modular representations lift naturally to $RG$, where the direct sum of the two (tensored with the fraction field of $R$) yields the same Brauer character.
In most cases for this $G$, having the same Brauer character implies having the same ordinary character. (This is easily seen by comparing the known values.) So this is a way of constructing distinct lattices which lead to decomposable and to indecomposable modular representations.
[I mention this particular group $G$ since it was already studied by Brauer and his student Nesbitt in the late 1930s. He must have realized that different lattices can give different modules for $G$, which probably stimulated his invention of "modular characters": traces of $p$-regular elements in $G$ when computed in $\mathbb{C}$ using lifted roots of unity. These were later dubbed "Brauer characters" by Curtis-Reiner, since their values are not actually "modular". Back in the 1980s I asked some experts such as Walter Feit and Jon Alperin about a question close to the question you raise and was told in effect that almost anything can happen; this unfortunately seems not to be discussed in the literature. I should have asked Brauer himself in the early 1970s when I got acquainted with him, but it didn't occur to me then.]
ADDED: I should also say something about the follow-up question, though it may be rather open-ended. What I mentioned briefly in my comment was a condition on the dimension of an irreducible module (over $\mathbb{C}$) which forces it to remain irreducible modulo a prime $p$. This comes from a result of Brauer which has a more general hypothesis not directly relevant to finite groups of Lie type when $p$ is the defining characteristic but satisfied sometimes by these and other groups (for various choices of $p$): the dimension of the given module is only required to be divisible by the power of $p$ dividing $|G|$. Beyond this I don't know any criterion forcing all reductions mod $p$ to behave the same relative to the decomposable/indecomposable dichotomy.
P.S. A further update is that it may be impossible now to track the intention of Curtis and Reiner in their exercise (since the latter was closer to the lattice questions and is deceased). But a possibly more elementary alternative to looking at groups of Lie type has meanwhile been suggested to me: consider the treatment of *Specht modules* for S_n by Gordon James, in Lect. Notes in Math. 682 (1978). Combine 12.2 with 8.11 (and the self-duality of simple modules in 11.5), along with the observation that a dual lattice yields the dual module after reduction mod $p$.