In short, no you can't merge files with-headers and without-headers. All must be one or the other.
So, the next step is to alter them so they are the same!
A lookup activity can get just the first row. If you know what the headers look like, then use an if-condition activity to check whether the returned row looks like a header. Then a copy activity can remove the header by using 2 similar datasets, one with "first row as header" (source) and one without "first row as header" (sink). This re-writes the file in place.
Once all the files processed, we use a copy activity to merge them.