Common problem usually addressed mechanically with gravity operated dampers or louvers to prevent back flow, but those too would be problematic on reversible fans.
The effect is that it does not actually change the magnitude of the starting current, but it does affect the acceleration time, often to the point where you exceed the thermal damage curve of the motor.
In general, transportation tunnel fan systems have some extra stringent requirements with regard to system reliability / redundancy, similar to Fire Pump controls, likely because of similar reasons. They are reversible not because of the train direction, but because in the event of a fire in the tunnel, the fans are operated to feed the fire, but isolate it in order to avoid having the entire tunnel become a blast furnace. So all of the fans on either side of an accident with incendiary possibilities are made to feed air TOWARD the area, and the fans nearest to it are made to exhaust out of the vents, which turns the local area into a small, but very controlled, burn zone. Side note: keep that in mind when driving in a long ventilated tunnel; if you are in or near an accident and fuel spills, GET THE HECK OUT!! They will sacrifice you to save everyone else.
OK, back to the controllers. I have done several of these systems, and besides the onerous process of NTSB investigation and certification, it is possible to do this with a VFD with Dynamic Braking, or a Soft Starter that can do DC Injection Braking. The control concept is in fact called an "Anti-Windmilling" start procedure. What you do is that whenever there is a "Start" command, in either direction, the first thing that happens is that the electronic brake is turned on and the motor shaft is stopped. Most electronic braking options will include a way to detect that the shaft has stopped to turn off that feature, so when it is done, the signal telling you that is then used to start the motor in the selected direction. The reason why this is not done with mechanical brakes by the way is because if the possibility of the failure of a brake coil creating a situation wherein the fan does not start.
Now if you must do this with an Across-the-Line starter (really? 350HP A-T-L?), then I suppose you could do it with a stand-alone DCIB unit attached to the starter. The thing is, it would need some sort of brains to make it all work, which implies either a bunch of relays or a small PLC, and getting that all certified to the NTSB reliability requirements may be problematic for a one-off project. Good luck.
