The living room in my basement is 12' 6" wide and roughly 25' long. The contractor I've talked to wants to seam on a 6" strip of carpet to make the carpet fit. Is this recommended? Or should I have them roll the carpet out and end up with a 1' piece at the end of the room? I don't feel like either option is a great option, but I don't know enough about carpet to make this decision.
As long as the add on is more than 1.5–2 inches wide, it will make a fine extension. Much narrower than that and the seaming iron and seaming glue will make a bit of a mess. Someone very experienced can probably manage it well enough though to make it turn out all right. The six inch piece will be invisible if professionally done.
If you have any concerns, have a reputable installer or have a written agreement that they will do whatever it takes to get the job done to your satisfaction.