Nearby blue compact dwarf galaxies (BCDs) are considered analogs to objects from the Epoch of Reionization revealed by JWST, having similarly low stellar masses, low metallicities, and high specific star formation rates. Thus, they represent ideal local laboratories for detailed multiwavelength studies of their properties and mechanisms that shape them. We report the first JWST MIRI/MRS observations of the BCD SBS 0335-052 E, analyzing MIR emission lines tracing different levels of ionization (e.g., [Ne ii], [S iv], [Ne iii], [O iv], [Ne v]) of the ionized gas. SBS 0335-052 E MIR emission is characterized by a bright point source, located in one of the youngest and most embedded stellar clusters (t ∼ 3 Myr, A_V ∼ 15), and underlying extended high-ionization emission (i.e., [O iv] and [Ne v]) from the surroundings of the older and less dusty stellar clusters (t < 20 Myr, A_V ∼ 8). From a comparison with state-of-the-art models, we can exclude shocks, X-ray binaries, and old stellar populations as the main sources of ionization. Interestingly, a 4%–8% contribution of a ∼10⁵ M_⊙intermediate massive black hole (IMBH) is needed to justify the strong [Ne v]/[Ne ii] and would be consistent with optical/UV line ratios from previous studies. However, even IMBH models cannot explain the strongest [O iv]/[Ne iii]. Also, star-forming models (regardless of including X-ray binaries) struggle to reproduce even the lower ionization line ratios (e.g., [S iv]/[Ne ii]) typically observed in BCDs. Overall, while current models suggest the need to account for an accreting IMBH in this high-z analog, limitations still exist in predicting high-ionization emission lines (I.P. > 54 eV) when modeling these low-metallicity environments, and thus other sources of ionization cannot be fully ruled out.