Migrate legacy application to new db/schema
As mentioned in the comments, this rather misses a question regarding a database technology. However here is my suggestion:

Since there are likely not any MySQL or Postgres specific features involved, just dump to CSV roughly in the data format that you will likely want in your target system and insert it there into any kind of table. If you afterwards need to massage data and add postgres features on top of it, create a new target table with the functionality you require (e.g. indexes, constraints, relations, views) and insert into it from the previous table.

I would not suggest anything more complicated, since you seem to migrate only small data.

PG has all the manipulation functions you need. Once you are done make it work in your application software.