We study generalized non-intersection probabilities for the three-dimensional Brownian loop soup at subcritical intensities. We establish the existence of generalized intersection exponents (GIE) and prove an up-to-constants estimate for these probabilities by means of a separation lemma tailored to this setting. We also relate the Hausdorff dimension of the set of local cut points of the three-dimensional Brownian loop soup to the GIE, and show that the GIE is continuous at intensity zero, where it reduces to the classical Brownian intersection exponent. In particular, this implies that, for sufficiently small intensity parameters, the set of local cut points has Hausdorff dimension strictly larger than $1$.