Context: This paper is part of a larger project in which we study the chemical abundances of extra-galactic post-AGB stars with the ultimate goal of improving our knowledge of the poorly understood AGB third dredge-up mixing processes and s-process nucleosynthesis. Aims: In this paper, we study two carefully selected post-AGB stars in the LMC. The combination of favourable atmospheric parameters for detailed abundance studies and their known distances make these objects ideal probes of the internal AGB third dredge-up and s-process nucleosynthesis in that they provide observational constraints for theoretical AGB models. Methods: We use high-resolution optical UVES spectra to determine accurate stellar parameters and perform detailed elemental abundance studies. Additionally, we use available photometric data to construct SEDs for reddening and luminosity determinations. We then estimate initial masses from theoretical post-AGB tracks. Results: Both stars show extreme s-process enrichment associated with relatively low C/O ratios of about 1.3. We could only derive upper limits of the lead (Pb) abundance which indicate no strong Pb overabundances with respect to other s-elements. Comparison with theoretical post-AGB evolutionary tracks in the HR-diagram reveals that both stars have low initial masses between 1.0 and 1.5 Msun. Conclusion: This study adds to the results obtained so far on a very limited number of s-process enriched post-AGB stars in the Magellanic Clouds. We find an increasing discrepancy between observed and predicted Pb abundances towards lower metallicities for all studied Magellanic Cloud post-AGB stars found so far, as well as moderate C/O ratios. We find that all s-process rich post-AGB stars in the LMC and SMC studied so far, cluster in the same region of the HR-diagram and are associated with low-mass stars with a low metallicity on average.