I have plans to supervise engineering work in Oklahoma and Texas in the near future. Will be travelling from Kentucky to above mentioned states. Are there problems with carrying an approved SBR from state to state as with Class III weapons? Do I need to seek additional paperwork or just leave weapon at home and take something else for SHTF weapon?

A Short Barreled Rifle is an NFA weapon, you need to file a from 5320.20 to move the gun out of state for any reason.

Quoted: A Short Barreled Rifle is an NFA weapon, you need to file a from 5320.20 to move the gun out of state for any reason.

True!

here's the linky-poo for the form

True you need the 5320.20, unless it's a C&R gun, then all you need is a copy of your C&R FFL.

Be careful about what you're doing.  The Form 5320 wants exact locations in the destination states and asks for specific reason for travel with the gun (i.e. gun show, permanent change of residence, etc).  The 5320 does not give you permission to take your SBR on further spontaneous travels anywhere you want to go.  The travel must be purposeful and reasonably exact.  If your 5320 lists "Dallas" as the Texas destination (and you aren't moving there), you don't want to be caught in El Paso with your SBR (after all, you have an NFA firearm with you, are not a Texas resident and you told NFA Branch under penalty of perjury that you were going to Dallas which is a long way away and not ON the way).  You also are expected to travel TO your 5320-listed destination with reasonable directness and haste, meaning going from Kentucky to Texas on an approved 5320, you do not want to stop too long in Tennessee.  Or stopping spontaneously to see your brother in Arkansas could get you in trouble with both the Arkansas authorities AND ATF.  

The McClure Volkmer Firearm Owners Protection Act (the very same legislation that banned further civilian MGs in 1986) gives you protection when traveling with firearms through states where they might be illegal, but that protection stops when you stray too far or for too long from the most reasonably direct route though the state.  When you stop to see the sights, you are no longer "passing through" that state en route to your destination and McClure-Volkmer disappears.

All of this is sort of moot if you fly as that is about as direct as it gets.

Anyway, I'm not sure you should be taking your SBR with you to OK and TX for a  "just to have it with you" or SHTF type reasons.  There's a lot to consider with NFA including maintaining "control" of it, not being caught in a position where someone else might inadvertantly but actually be in "possession" of it if you left it somewhere to go eat dinner or grab a beer (friends, family members who are staying with you or you with them, etc).

With Title I guns, you aren't bound by the same travel or possession regulations and you'll have far greater freedom to change your plans at the last minute.  

Just slap a 16" upper on it and call it good (if an AR).  Too much of a hassle to transport C3.

If it's NFA registered, it's still an SBR even with a 16" or 20" barrel.  All the NFA rules apply even if you don't have the actual short barrel on it.

How long does it take to get the approved form back from ATF?