Good news; finally found the straight answer from the Texas CHL website.
Q: Can a license holder have a handgun in their vehicle in a school parking lot?
A: A license holder is not prohibited from having a handgun in his or her vehicle in a school parking lot if the parking lot is not posted with the 30.06 signage described above.
(School employees should know and comply with their employer’s policy on this point.)
It is, however, a criminal offense for any person who is on school property to exhibit, use, or threaten to exhibit or use a firearm.
See Texas Education Code § 37.125(a).
There are no 30.06 signs in any school parking lots in the district. Therefore: Lockbox under the seat, say nothing to anyone, and I should be in the clear whether it's detected or not -- and from what I've read here, it doesn't sound like that's likely anyway.
I'll be carrying the stick anyway. Thanks to everyone!