You have what you need. There is a button on the front of the board, so as long as the board is accessible to be a button for you near the gate, you can just use that to turn on the relay to cause an event that an automation rule can use as a trigger. "Switch" should be the lower button, to toggle on and off.
Press the button on the front of the device to start the automation timer for the pulse to happen after 50 minutes. This will leave the lock solenoid powered, which is not what you want, so the first thing the rule would would do is turn the relay back off, and then it would start a timer to run in 50 minutes to to do the on-off pulse. "Off" is a separate event, so that would not start the timer, when you turn the relay back off.
That gives you a button next to the gate that starts the timer, without adding another device. For the automation, you can use Simple Rules, Rule Machine, Webcore, or maybe a custom app. I haven't tried visual rule builder, but if you are good with flowcharts, that may be something to try.
Using the board button as the trigger event means you have to select a "physical" event for the trigger event, as that identifies you as physically pushing the button on the board. A "digital" event is when an automation turns the switch on. So, you don't want your rule to react to the 50 minute timer pulse when the automaton turns the relay on, as that will start another 50 minute timer, so it would just pulse every 50 minutes.
The rule is just one "if" condition, in Rule Machine the relay state change is called a trigger, and it triggers something to happen, or a list of things happen.
If the latch switch turns on, and the event is physical
turn off the latch switch ** so it doesn't stay powered
wait 50 minutes
turn on the latch switch
wait 2 seconds
turn off the latch switch
You may want to check out this custom flowchart type automations app: