In the scope of the Physics Beyond Colliders studies, the Gamma-Factory initiative proposes the use of partially stripped ions as a driver of a new type of high-intensity photon source in CERN Large Hadron Collider (LHC). In 2018, the LHC accelerated and stored partially stripped 208 Pb 81+ ions for the first time. The collimation system efficiency recorded during this test was found to be prohibitively low, so that only a very low-intensity beam could be stored without the risk of triggering a beam dump when regular, minor beam losses occur. The worst losses were localised in the dispersion suppressor of the betatron-cleaning insertion. This article presents an analysis to understand in detail the source of these losses. Based on this understanding, possible mitigation measures that could significantly improve the cleaning efficiency and enable regular operation with partially-stripped ions in the future are developed.