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Both Kansas and Missouri have constitutional carry, so if those are the only two states you will visit you are probably gtg.. (I think) I am a Missouri resident and we stayed at a hotel in Kansas for a while due to home damage, and the way I read it, I did not have to be a Kansas resident to constitutional carry. Only legally able to posses a gun. I have not read up on if a non resident can constitutional carry in Missouri.
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I took a quick look at Missouri's CCW statutes and it appears that for a MO permit, one must be resident of MO or a member of the armed forces. I didn't see any provisions for a non-resident permit.
http://www.moga.mo.gov/mostatutes/stathtml/57100001011.html
For most purposes, I believe OP could rely on MO's constitutional carry. He specifically mentioned school zones, though, which do require a permit from the state you are in. I've never heard of anyone getting in trouble for that and practically everyone who has carried out of state has passed through school zones. It does seem like a good idea to get an in state permit if you're going to do that frequently, but I don't know of any way that OP can.
It is probably worth while to get some kind of permit that Missouri will recognize, though, because of the off-limits places. With a permit, one is subject to removal but it is not a criminal act to carry in these places. Without a permit it is a misdemeanor. See below for a list of the off-limits places. Be sure to read subsection 2 near the bottom.
http://www.moga.mo.gov/mostatutes/stathtml/57100001071.html
Oh, and congrats on escaping California, OP!