I think that this warning was in relation to a system designed to prevent runway overruns...which we don't have, so I don't know the details.
In our operation, we'll have a look at the landing data on the OIT (the laptops). Using the 'dispatch' module, and the actual conditions, gives a runway length requirement of about 2450 metres. That includes a very large margin...but still gives around 500 metres excess. Using the 'flight' module uses less margin, and gives much more spare. Obviously the runway was perfectly acceptable.
Our FMCs don't get into the act, other than providing the navigation, laterally and vertically. The performance/approach page will have our selections of flap and approach speed, and will also include the ATIS data. It doesn't do any runway length calculations.
I presume that the overrun system has the ability to dynamically monitor the performance...so that it can take into account excess airspeed, or adverse wind. Otherwise it would have little value.
As for the response..the first go around would be in accordance with the SOP. After that, having looked at the OIT again, and being aware that it's a normal runway used every day...I'd land and u/s the system. I expect that when I'm replaced by a robot, that it will continue to go around until the fuel state becomes an emergency, at which point that would leap to the top of the priority list and allow a landing.