Context. As some of the oldest stellar systems in the Universe, globular clusters (GCs) are key fossil tracers of galaxy formation and interaction histories. This paper is part of the LEWIS project, which provides the first homogeneous MUSE integral-field spectroscopic survey of a complete sample of ultra-diffuse galaxies (UDGs) in the Hydra I cluster. Aims. We use MUSE spectroscopy and new VIRCAM H-band imaging data to study the GC populations and dark matter content in four dwarf galaxies from the LEWIS sample, which were found to host several GC candidates based on previous photometric studies. Methods. We retrieved line-of-sight velocities (LOSVs) for all the sources in the observed MUSE fields and classified them based on their spectral features and LOSVs. Because the spectroscopic measurements are limited to relatively bright sources (mH ≲ 23.5 AB mag), we developed a multi-band photometric procedure to identify additional GC candidates that are too faint for spectroscopic confirmation. GC candidates were selected based on a combination of photometric properties (colors, magnitudes) and morphometric criteria (shape and size). The same selection criteria were applied to empty fields to estimate a statistical background correction for the number of identified GC candidates. Additionally, H-band observations were used to constrain the stellar masses of the studied galaxies. Results. Based on the spectroscopic classification, we confirm one GC in UDG 3, two in UDG 7, and four in UDG 11, while UDG 9 has no spectroscopically confirmed bright GCs. We identify four intracluster GCs in the vicinity of UDG 3 and UDG 11, and one ultra-compact dwarf (UCD) with a radial velocity only −85 ± 10 km/s different from that of UDG 7 and thus possibly bound to it. Considering the completeness correction and accounting for possible contamination by unresolved background galaxies, from the photometry we estimate that the number of GCs ranges between 0 and ∼40 for the investigated UDGs. Their specific frequencies suggest that three out of four UDGs are either GC rich, similar to those in the Coma cluster, or belong to an intermediate population, as seen in the Perseus cluster. Dark matter content estimates, inferred from GC counts and stellar mass, indicate that these galaxies are dark matter dominated, with dynamical-to-stellar mass ratios ranging from ∼10 − 1000.