Alrighty, after much research, got it working. Will list all the settings in case anyone with the same setup is having issues.

Clarus 500 GC
Turbomatrix 40 Headspace Sampler (pressure-balanced)

Clarus 500 Settings:

Column: 30m x 0.53mm id x 3 micron film of DB-624
Zero dilution outer/inner split/splitless liners used instead of standard 4mm id split liner
Injector, detector, and oven temperature programs as per USP
Carrier flow: 3 ml/min He
Split: 5:1
Hydrogen flow: 45 ml/min
Air flow: 450 ml/min

Turbomatrix 40 Headspace Settings:

Carrier Pressure: 48 psi
Oven Temp: 80
Needle Temp: 105
Transfer Temp: 110
Thermostat Time: 20
Pressurize Time: 2.0
Inject Time: 0.12
Withdraw Time: 0.1
Vial Venting: On
Operating Mode: Constant
Injection Mode: Time
Transfer line: deactivated fused silica line 0.32 mm id

Chromatography is beautiful...even better than the sample chromatograms I had on hand. As far as I can see, the biggest changes that made it function properly were the installation of the zero dilution liners, use of the bigger id column (though I suspect that my 0.32 mm id column might just have gone bad), and the headspace carrier pressure increase to 48 psi to ensure maximal headspace transfer.