Can I add "Included Columns" to an index without affecting performance?
Why would you do anything through the SSMS GUI???

Try doing it as a pro, by a script. Your best option is to create the index online with drop existing, changing the definition to match the new desired schema in the process:

create index ... on ... include ... with (online = on, drop_existing = on);


This, obviously, is subject to the usual restrictions for ONLINE operations, and will cause the normal overhead caused by an ONLINE index create/rebuild operation. But the good news is that the operation will be online, will not block existing queries and the old index will be available for queries while the new one is being built. There will be a query drain stage at the end when all queries will block while the indexes are being swapped, but that should be a very fast operation (as long as there are not long running user transactions blocking the swap itself).