I personally think that rather than fear of the hero causing a new war, it's probably because the hero "righteousness" which from the standpoint of the king is a tad bit too much to handle, which in turn would probably become the cause of many conflict with his own gains and goals.
If the king himself is indeed that kind of person as described, rather than fear of his kingdom being attacked, he would fear that the hero would just mess with his plans, cause a revolt and would eventually overthrow him and his own government because of her sense of righteousness while also gaining the citizen support for it. (that in itself is a threat to the kingdom, more precisely to the current head of state)
As a precaution for that, killing the hero would be one of the appropriate measures before she started doing just that.
Other means would be making a "puppet" out of her.
It is logical to handle any kind of possible threat by means necessary.
And, as for the other head of state... The hero would just indeed become a trouble for them and the world ahead if she is indeed that kind of person. As much as I hate that kind of notion, a single individual with that much power in the whole world would indeed be nothing but a troublesome matter, I think even the hero understand that.( tho' being fine with being executed is a bit too much I think.)
As for why I don't think it's because the hero would start a war... war itself is pretty much contradictory to peace that the hero long for... a revolt is another thing. From the look of it the hero may still be okay with causing a revolt for a good cause, but I don't think she would cause a full scale war. (then again, it's the person who is okay with being executed...)