Reasoning for Selecting Barrett only at the INTT Stage?

Hello,

I was wondering why you decided to use Montgomery in both of the NTT and INTT as well as including Barrett in the INTT only. From my understanding Barrett could be used in the NTT and INTT with no need for Montgomery. I'm sure there is a valid reason but if you guys could enlighten me as to why I'd really appreciate it.

Thanks!