I have a very similar situation, there are a couple cattle gates that I don't want the dogs crossing, even if they are open. I was "on the fence" 😆 so to speak. Invisible fence was the solution. I don't have "door dashers" but if you open a gate long enough to get a tractor through , by the time you walk back to close it a curious dog will go through. Now the doggies thing the fence shoots lightning and they know it's "range". It's not as gentle as I'd like to be, I really am one foot in on any type of Ecollar, but a lightning bolt to the neck is a lot nicer than letting fido pick his boundaries and having him get run over or shot by a trespassing hunter.
Oh, most importantly. This method works with no treats. I will not do treat work. AND this method requires no commands. A dog cannot hear "sit" , "stay", or any verbal command while a tractor is thundering through. My dogs learned quickly that the only time they go through the gate is when a leash is on, and a sit,stay,and break are issued.