I don’t understand why it is allowed to treat the red expressions equally. I mean, there are 2 countries… Then, the red expressions are obviously not the same (?) because:

I use a similar model like in Sims’ notes: https://www3.nd.edu/~esims1/new_keynesian_2017.pdf.
On page 5 below, you see the standard NK intermediate firms’ FOC. Nonetheless, this is not a 2-country setting. CPI and PPI are identical and multiplication is obviously allowed (blue terms).

In a 2-country model, the multiplication (without differentiation) is in my opinion not allowed, as the CPI in each country is an aggregator of domestic as well as of foreign PPI’s. These variables are not the same before steady-state?

I think the multiplication is allowed. you need to convert everything to the same currency using the NER and divide by the CPI and then evaluate in utility terms. In ss, you will get the standard price= markup*nominal mc relationship in closed as well as open economy models. No problems there.
I have put some notes for calvo pricing in an SOE (not 2 country) model at https://sites.google.com/site/reubenpunnoosejacob/dsge-technical-notes