One thing I've been told by many sailors is that when the outside boat calls for no room and you think it's close you should turn toward them immediately and if you touch then you should protest because you had overlap.
I don't agree with this approach because you should always avoid contact but I've been told this by many seasoned sailors on big boats.
Turning down is pretty much your only option unless you are planning to do a peal at the mark. You are not turning down to force the issue and make contact you are turning down because you cant't stay inside and the outside boat said you were clear.
