In JOSM you can do this easily, with OOTB tools:
- Download a small part of the map where the busline goes with the down button.
- Enable the relation list window: Alt+Shift+R, or there is a button for it on the left vertical toolbar, there is a chance, that it's enabled by default.
- Find the relation in this relation window, you can use search there. Search for the name or the ID of the relation.
- Rightclick on the relation in this window: Download members, then rightclick again -> Duplicate
- After confirmation it will open the relation editor window. It's not hard to understand, there is a nice tutorial about it on the josm wiki: https://josm.openstreetmap.de/wiki/Help/Dialog/RelationEditor You will have to select lines and add them with the arrows. It will show if there is a gap in the path.You can reorder already added streets by drag and drop.
- If you finished OK, than Upload!
JOSM seems like a beast at first, but once you learn to use it you can't go back to iD, it's annoyingly inefficient. Don't try to learn everything at first, start slowly, one feature at a time. I still don't know everything after 747 changesets from JOSM. RTFM, everything is clearly written in the osm and josm wikis. Anywhere in the program you hit F1 it will open a contextual help.