You've exposed your tank to cyano by having it within 3 miles of a body of water. I'm only half kidding. You should probably rinse the cyano off, or see if you can get it off with a paper towel, but it's not like it's a "bacterial disease". It is, but it's a disease in the sense that the common cold is a disease, but it's almost always ever-present in every large population. Your tank is a large population of bacteria as it is. But you're seeing it on the macro level, not the micro level. It's already in your tank, waiting for the appropriate excess nutrients (namely higher phosphate with low nitrate) for it to show up on it's own.
In theory the plug could have high phosphates, and could leach them back into your tank if your tank has low phosphates, causing algae and/or cyano, but that growth wouldn't be from the cyano, but from the phosphates bound by materials in the plug.
Short answer is you won't be exposing your tank to anything it doesn't already have by adding the frag.