[SalesForce] Receiving “Profile does not have access to” in the flow when calling invocable apex class

I am confused as to what you are really trying to accomplish with this structure. You don't need to create any logic for the actual order of execution (which, correct me if I am wrong, it appears you are trying to do here). Triggers have a very specific order of execution. It will actually run through your trigger several times to hit before and after logic.

Wouldn't it be much easier to do something like:

trigger OpportunityTrigger on Opportunity (after insert, after update, before delete) {
    if(Trigger.isAfter && Trigger.isInsert){
        OpportunityHandler.createScheduleLines(Trigger.newMap);
    }

    if(Trigger.isAfter && Trigger.isUpdate){
        OpportunityHandler.adjustScheduleLines(Trigger.newMap, Trigger.oldMap);
    }

    if(Trigger.isBefore && Trigger.isDelete){
        OpportunityHandler.removeScheduleLines(Trigger.oldMap);
    }

}

public class OpportunityHandler{

    public static void createScheduleLines(Map<Id, Opportunity> newMap) {
        // logic
    }

    public static void adjustScheduleLines(Map<Id, Opportunity> newMap, Map<Id, Opportunity> oldMap) {
        // logic
    }

    public static void removeScheduleLines(Map<Id, Opportunity> oldMap) {
        // logic
    }

}

Let Salesforce worry about the order they are executed.

When an InvocableMethod runs, it runs against all interviews in a batch at once. As such, the number of outputs, if any, must match the number of inputs.

In order to fix this error, you need to pass in some value so you can return a list of values all at once.

Something more appropriate would look like:

static Set<Id> recordsProcessed = new Set<Id>();
@InvocableMethod static Boolean[] runOnce(Id[] recordIds) {
  Boolean[] results = new Boolean[0];
  for(Id recordId: recordIds) {
    results.add(recordsProcessed.add(recordId));
  }
  return results;
}

What is happening here is that we call the Set.add method to add the record Id to the Set. If the value already exists in the Set, this method returns false, meaning that the record has already been processed once. If the value does not exist in the Set, the method returns true, meaning that the record has not yet been processed.