The problem with orcs, of course, is that they're orcs. That's the problem that requires a solution. You can't fix them, so they have to simply be removed from a society, preferably by the most permanent means available so you're not dealing with the same problem next week. They breed like cockroaches.
They were Orcs when they settled into the land in the first place, was it a problem then?
The previously mentioned ideas of sending them out into the military, or finding an appeal to have them move, makes use of their strengths. An Orc may make a terrible gardener, but a great bouncer. Put them where they would be most useful.
Does the kingdom have a border territory with a rival nation or a dangerous wilderness? Could the king find a way to make it appealing for the Orcs to move there (Free land for Orcs, or a tax break, or great game hunting, whatever works) something appealing enough for the majority to move away from the rest of the population.
If the Orcs are an overt problem to the rest of the population, why can't the King use hostile, overt methods? If he's have the support of the people, than he can get away with genocide (disclaimer: I'm not condoning genocide) . If he'd lose favor with the people for overtly eliminating the Orcs, then perhaps they're not as bad of a problem as the King things they are?
I really like your dilemma. Could you tell us more about the delicate political situation, and the possible civil war that might erupt? I think a better understanding of "What would happen if..." any of the methods you've suggested occurred would make it easier to offer suggestions.
To offer up my own alternative ideas:
-Start a new Orc religion. If a charismatic Orc could be inspired to lead a new religious movement that was Orc focused and emphasized peace or a certain way to channel thier feelings of hostility, there might be enough converts to alter the state of "Orc-ness". 5 levels of the PrC Evangelist from Complete Divine plus some goodies from Book of Exalted Deeds should do it.
-Build a combat Coliseum. Give the Orcs (and everybody else) a venue to channel aggression in a way that is constructive/entertaining/legal. Design the law so that if anyone genuinely has a feud with someone else, they must act in out publicly. This will turn all tavern brawls into spectator-witnessed entertainment gladiatorial events. Have strict regulations on how the Coliseum is run, so it doesn't become a training ground for a military uprising.
-Is the kingdom anywhere near water? Develop a Navy dedicated to war/exploration/something. Find a way to advertise Orc seamen as valuable. Locate an island or some other territory far away. See that the Orcs that land there end up with some legal claim to the territory according to your laws, even if someone else is already settled there and you'd need to go to war with them. Advertize this at home as a very Orc thing, and that territory should be, must be taken in the name of Orc pride.
A few other questions-
-Do they have the Light Sensitivity as stated in the MM I? Is there a way to increase the brightness of the kingdom to make the area less tolerable? Shiny brass fixtures as the new trend,
Daylight streetlamps during the night time, open courtyard homes with large windows as the latest domestic design, etc.
-Do they have the extreme hatred of Elves and Dwarves as stated in the MM I? If the kingdom welcomed in more Elves and Dwarves peacefully, under the same gesture of peaceful acceptance that the kingdom exhibited when welcoming in Orcs, and the Orcs had objections, could the King eventually make a proclamation that "We as a kingdom are tolerant of everyone, if anyone within the kingdom has objections to this policy, they can leave. Either of their own free will, or if they cause trouble, they will be made to leave" Then it would be the Orcs own actions that trigger the conflict, not the King. Having gained an increased Elf and Dwarf population should endear the Kingdom to Elven and Dwarven allies, who could provide military aid, should it come to conflict. This idea assumes many things about the nearby kingdoms, and may not fit within your campaign world.