Strengths:
It is nice that you discovered a simplified model (with no leverage) did better. The maximiziation probably works a bit better for a simpler model, explaining the slightly higher likelihood obtained. https://ionides.github.io/531w22/final_project/project14/Blinded.html used a similar model and found a simpler pomp representation for it.
Suggestions:
It would be interesting to see if a longer-tailed distribution for \(\epsilon_n\), such as t, fits better.
The global search suggests a bimodality: a collection of searches seem to reach a different and inferior region of parameter space. This is not scientifically critical - the second mode has much lower likelihood - but may help to understand numerical issues.