Are you sure the stat branch line builds pressure on a call for heat. Normally the opposite is true. The branch line exhausts to drive the VAV controller to close to minimum for reheat airflow(The VAV controller will have a seperate supply main).
To verify the stat:
Start at the stat-is supply pressure sufficient to close the switches, if not problem is in the pneumatic supply not the stat.
Next check the stat output-disconnect the branch and connect an accurate gage. Turn setpoint to full cooling(on DA stat temp up pressure up)- gage should read supply pressure.
Next,if not, depends on the type of stat but most brands have replaceable arms. Old school actually... today most would just try a new stat.
If yes then verify tubing to VAV controller. Disconnect branch line at controller and repeat above steps....
Enough steps for now, Reply with how you make out.
24 Years Under My Belt,
Too Many Cheeseburgers Over My Belt,
But I Love To Talk Controls.