Take $G_m$ acting on itself via $z\mapsto z^2$, and take $k$ to be of characteristic $2$. The equivariant derived category here is the derived category of $\mathbb{Z}/2\mathbb{Z}$-modules. Now take the regular representation. It’s indecomposable. If you forget the equivariant structure it decomposes.
Note: this gives you something in the heart of your favorite t-structures that satisfies the requirements, or if you like you can use Yoneda Ext to produce a genuine complex in $D^b_{G_m}(G_m) = D^b_{\mathbb{Z}/2\mathbb{Z}}(pt)$ satisfying your requirements. Doesn’t contradict my remark about the forgetful functor being faithful on t-structures (since splitting has to do with existence of idempotents, rather than morphisms becoming $0$).