The whole file remains one polygon though. I've tried the Integrate tool but no success. I wonder which tool to use instead, if I am making a mistake regarding the reprojection or if there is any cosmetic fix to make the line disappear (but keep other outlines) in maps?
Normally I use this for a similar looking display issue East-West across the map, but you could try moving the wrap parameter the .prj file. (e.g.
+lon_wrap=-180) See wrap parameter explanation here.
I do this by creating a custom projection. Then you should be able to customize/write your own .prj file. See the documentation links for Arc and QGIS below.