We consider a class of Lévy-type processes with unbounded coefficients, arising as Doob $h$-transforms of Feynman-Kac type representations of non-local Schrödinger operators, where the function $h$ is chosen to be the ground state of such an operator. First, we show the existence of a càdlàg version of the so-obtained ground state-transformed processes. Next, we prove that they satisfy a related stochastic differential equation with jumps. Making use of this SDE, we then derive and prove the multifractal spectrum of local Hölder exponents of sample paths of ground state-transformed processes.