Hello, I received many valuable information about my question. The conclusion is that having a son called Chaim Josek and a father called Josek Lejb is not contrary to the Ashkenazic naming pattern because these are different names. But most of all, it seems like the 2nd part of the name is precisely there to remind the father's name, with the omission of "Ben" or "_owicz".
Thank you again (I think I personally thanked every contributor).
