OK, here are some words of wisdom from someone in your position in 2012 - most from others more expert who advised me beforehand. However, our Avion T20 is 2/3's your AS length & less than 1/2 the weight (3-3,500# wet/loaded), & I only towed 1/2 your trip in 800+/- miles from ABQ to OC, & it was in July - so no icy/snowy roads for me.
You'll have a monster of a 31' & probably more like 6-8000# wet/loaded "Silver Twinkie" in winter weather, so you may want to wait for summer (you don't say when you'll do this in your OP). You've also got a lot of moving pieces in the equation - pretty much all of them!
Before the TV which everyone has focused on, let's address the trailer, & I'm surprised nobody has brought this one up yet, but before you go anywhere with it .... get the AS slow & careful to a very local KS trailer shop (maybe the seller will take it there for you before you get there?), & have them do the following:
1. Get the wheel bearings checked & replace seals/bearings if needed & repacked; &
2. Check the brakes & repair if/as necessary, & adjust them, 7 check the electrical system from the plug for proper operation from the TV & 7-blade connector, & change to a modern 7-blade connector if it's still got the old style (which won't fit your new truck's 7-blade as noted in another post above); &
3. Have them double check all the lighting just replaced for proper operation, & do an overall full safety check on the AS for a long XC trip, as well as checking the safety chains (replace if necessary), & a new emergency break-away brake switch (unless it's recent/new already); &
4. Get an accurate weight for the trailer as wet/loaded for your trip + find out the total gross rated weight for it - so you know what TV & hitch to get with a proper safety margin; &
5. Have them mount & set-up ~ or check your mounting/set-up on the WD & anti-sway hitch of your choice (do some research on them before you decide) - & make sure they show you all of the WD set-up & adjustment, hitching & electrical hook-up procedures, & what/how to do the on the road checks.
On to the TV questions, you need to check with CA DMV on that (or AAA if you're with them) on the ins & outs of buying a vehicle out-of-state - especially new, because they make it difficult & assume it's a sales tax evasion ploy, & you could be stung by double state sales taxes on it.
Also, you'll need temp plates for both the AS & TV on the trip days - so also check with KS DMV beforehand, & then you'll need to re-register both TV/AS once in CA (almost immediately). I don't know all the process to do so properly, but I know horror stories of those who didn't get it done right!
For me, I got a NM 2 day temp paper plate for the trip & a bunch of their documentation, did the trip in the allotted time (NM's max temp BTW), & registered our Avion on the way home at Anaheim Hills AAA on the way to Orange (truck was rented, so not an issue for DMV).
Look at which Truck you'll need only after you have an actual GVWR on the 31' AS (IIRC AS has their old specs at their website) - dry wt. is meaningless for towing, since options, LP, water/gray/black tanks & all sorts of other stuff, plus whatever you put into the trailer, mods over the years, etc., etc. - will add to that dry weight.
If the seller could fill the tanks & take it to a CAT scale for you for the actual wet/loaded wt., that would be helpful.
Then you'll need a truck (or any other TV) rated for both the actual wet/loaded & max. trailer wt. & hitch wt., so that may be Ford's top level tow package on the F150, or perhaps an F250 ~ or maybe the best deal is on one or another if either could work. I know the base F150 is or was only rated to 5000# & 500# hitch wt. as of a few years ago, & then there were 1-3+/- levels of tow options with increasing ratings & cost.
Having rented the Ford F150s & F250 for TVs since getting our Avion, I can say they have a good built-in anti-sway system on the truck, but you'll also want that anti-sway hitch on the trailer too, with such a long trailer due to sail-effect from the side area. If you plan to get a WD/Anti-sway hitch anyway - then get it now for what may be it's longest trip in your ownership.
Also get the brake controller in the option package or aftermarket & installed & checked by the selling dealer (then it's their liability for any errors & rolled into the financing).
I'd also opt for the dual 4-pin/7-blade type receptacle on the truck for more flexibility, & a back-up camera to aid in hitching up - especially solo.
You'll also need to factor in your own needs, budget, how much you'll be towing it, where you'll keep it, etc. to decide what is best for you.
Hopefully we'll see y'all at one of the CA vintage trailer rallies & events in the near future. Get Vintage Camper Trailers magazine from their website for the dates on those mostly weekend events, & put in for them as early as possible cuz they fill up fast, but you can also put in for wait lists & often get in at the last week or so due to cancellations.
PS - make sure to have in the rig - a tool kit, jack sized for the trailer wt., wheels chocks, wheel lug wrench to fit the AS lugs, emergency triangles, flashlights, safety vest, emergency magnetic towing lights with enough 4-pin wire to react 35+/- feet to the hook-up, & spare parts & tires (TV & AS), etc.
.... & FYI, I got the Hensley Cub WD/Anti-sway hitch for our Avion (their Arrow or a ProPride would probably be needed for your 31' if it grosses over 6000# TW & 600# HW) - which are the top of the line & eliminate sway, with many other choices beyond those PPP hitches.
Good Luck from an OC vintage kin owner, & former SD resident (Uni High class of `70)!