Unlike a belt (which my Kingfisher uses) the chain should probably have a bit of slack, kind of like a well adjusted bicycle chain. Sounds like the FEC750 could use an adjustment roller (kind of like an idler pully) to allow fine tuning. I'd give CookShack a call.
Chain will stretch (unless it's high dollar S.S. pre-stretched) over time or as it heats up. Swapping for a 1/2 link may work today but the best idea is to use a tensioner that you can adjust or a spring loaded tensioner (more expensive). If you have no place to hard mount then you could use a floating tensioner. Plenty to choose from in Mcmaster-Carr.
I'd call the manufacturer first though and see if they already have a solution.