As your goal is still not clear to me, I cannot comment on which approach is correct. My hunch is neither. The paper you reference estimates the model on pre-Covid data because it assumes that the structure of the economy has stayed the same after 2019, only the shocks have become bigger. It then does IRFs keeping the parameters at their old value, but with a larger standard deviation for the shocks. That seems different from what you are trying to do.
- Option 1 would be forecasting data without any shocks occurring in the second period. That seems inconsistent with what you are trying.
- The point of full information estimation is to perfectly fit the observed data. After all, you assume the model is the data-generating process.
- and 4.: A good starting point is How to see whether the model fit the real data well? - #2 by jpfeifer