Gamma-ray bursts (GRBs) offer a powerful window to probe the progenitor systems responsible for the formation of heavy elements through the rapid neutron capture (r-) process, thanks to their exceptional luminosity, which allows them to be observed across vast cosmic distances. GRB 241105A, observed at a redshift of $z = 2.681$, features a short initial spike ($\sim$1.5 s) and a prolonged weak emission lasting about 64 s, positioning it as a candidate for a compact binary merger and potentially marking it as the most distant merger-driven GRB observed to date. However, the emerging ambiguity in GRB classification necessitates further investigation into the burst's true nature. Prompt emission analyses, such as hardness ratio, spectral lag, and minimum variability time-scales, yield mixed classifications, while machine-learning-based clustering places GRB 241105A near both long-duration mergers and collapsar GRBs. We conducted observations using the James Webb Space Telescope (JWST) to search for a potential supernova counterpart. Although no conclusive evidence was found for a supernova, the host galaxy's properties derived from the JWST observations suggest active star formation with low metallicity, and a sub-kpc offset of the afterglow from the host, which appears broadly consistent with a collapsar origin. Nevertheless, a compact binary merger origin cannot be ruled out, as the burst may plausibly arise from a fast progenitor channel. This would have important implications for heavy element enrichment in the early Universe.