My father worked as a longshoreman at the port of Montreal when he came back from the war. It was one of his first jobs, because it was easy to find work there at the time. The system was much the same: he had to go to the docks and if there was work for him, he would be hired. If there was no work, he would come home. That was in 1945. A long time ago.
I am wondering whether there are ports in other parts of Canada where things are different. Is this how things work in all ports? Did you experiment it in other places? Did people from the union and the employers hold discussions to try to come to an agreement that would facilitate things?
If people were to use the Internet, I would be afraid that there might be discrimination because you can see a person's name. Perhaps you could do this by phone. If the union were to call people individually, rather than ask them to go somewhere, it might be preferable. I think that is what is being done currently in Montreal. Have you tried to work differently and to use other methods of facilitating the involvement of women and men? Generally, when you make things easier for women, it makes things easier for men as well.