So, I ended up figuring this out and wanted to post the resolution....It was actually a simple fix, but tricky to discover and I wouldn't have done so without some fabulous tips from Ashish in Premiere support = #rockstar!

Because we have workflow rules that update opportunities, that was what was triggering the double post. To work around this, I added a class:

global class test1{
public static boolean flag=true;
}

Then, reference that class in the trigger so it doesn't post twice:

if(test1.flag == true){
test1.flag=false;

The full code of the trigger is:

trigger AddClosedWontoChatter on Opportunity (after update) 
{
    //Setup the Feeditem array of items to post to Chatter
   list<FeedItem> fplist = new list<FeedItem>();
    
    //Setup the array to hold the ids to Iterate through
    Set<Id> pbeIds = new Set<Id>();
    
    //Iterate through the Opp
    for (Opportunity o : Trigger.new) 
    {   
       // Create individual post
       pbeIds.add(o.Id); 
    }
    
    //Setup APEX MAP Arrays to get the required data from the Pricebook and Opportunity 
     Map<Id, Opportunity> Opp = new Map<Id, Opportunity>([select Account.Name, Account.ID, Owner.Name, Owner.Id, Owner.UserRoleId from Opportunity where id in :pbeIds]);
    
    //Iterate through the line items once again to add the data to the Chatter Object
    for (Opportunity o : Trigger.new)
    {
           Opportunity oldo = Trigger.oldMap.get (o.id);
           //If the part outcomes are different, let's add to the chatter feed
              if(o.StageName != oldo.StageName && o.StageName == 'Closed Won' && o.Account.Id != '0013000000GJRrU' 
&& ((o.Invoice_Closed_Won__c > 79999 && (o.Type == 'Affiliated New' || o.Type == 'Renewal'))
|| (o.Invoice_Closed_Won__c > 19999 && (o.Type == 'SMB - Affiliated New' || o.Type == 'SMB - Renewal' || o.Type == 'SMB'))
|| (o.Invoice_Closed_Won__c > 29999 && o.Type == 'New')
|| (o.Invoice_Closed_Won__c > 5999 && o.Owner.UserRoleId == '00E30000001U9dG' ))
)      
           {
     if(test1.flag == true){
    test1.flag=false;

    FeedItem fpost= new FeedItem();  

               fpost.parentId = '0F930000000CbQL';
               fpost.createdbyid = UserInfo.getUserId();                              
               fpost.Type = 'LinkPost';
               fpost.Title = o.Name + ' - $' + o.Invoice_Closed_Won__c;
               fpost.Body = 'has Closed Won the following $' + o.Invoice_Closed_Won__c + ' opportunity on Account: ' +opp.get(o.Id).Account.Name+ '. Congrats, ' + opp.get(o.Id).Owner.Name + '!'
               + '\n*Application: ' + o.Application__c
               + '\n*Reason Won Details: ' + o.Reason_Won_Lost_Additional_Details__c;
               String id = String.valueOf(o.Id).substring(0,15);
               fpost.LinkURL = 'https://na1.salesforce.com/' + o.id;
               fplist.add(fpost); 
        }   
       } 
       }    
              
    //if the post is empty, don't try and post it to Chatter    
    if(!fplist.isEmpty())
    {
        insert fplist;
    }    
    

The test class is:

@isTest
public with sharing class ChatterClosedWonOppTests {
    /*
    *    Test the AddClosedWontoChatter trigger which inserts chatter feed posts whenever an opportunity is closed won at a certain value
    */
    public static testMethod void testAddClosedWontoChatter() {
    
        Profile p = [select id from profile where name='Standard User'];

        User u = new User(alias = 'test123', email='test123@noemail.com',
            emailencodingkey='UTF-8', languagelocalekey='en_US',
            localesidkey='en_US', profileid = p.Id, country='United States',
            FirstName='Test',
            LastName='User',
            timezonesidkey='America/Los_Angeles', username='test123@noemail.com');
        insert u;
          
        CollaborationGroup c = new CollaborationGroup(name='Test Group',CollaborationType='Public');
        insert c;
        
        //create the test opportunity
        Opportunity testOpp = new Opportunity( Name = 'My Test Opp',
        StageName = '2 - Discovery',
        Owner = u,
        Type = 'SMB',
        Invoice_Forecast__c = 25000,
        CloseDate = System.today());
        insert testOpp;   

        //Since it is an after update trigger, update the opp that was just inserted.
        testopp.StageName = 'Closed Won';
        testopp.Reason_Won__c = 'Price';
        testopp.Competition__c = 'None';
        testopp.Application__c = 'Compliance';
        update testopp; // this statement will fire the trigger 

// Get the feed post that was supposed to be created based on
// closing the opportunity
    
  //  List <FeedItem> posts = [SELECT id
   // FROM FeedItem
    //    WHERE ParentId = :c.Id];
  //  System.assertEquals(1,posts.size());

   //    List<CollaborationGroupFeed> posts = [SELECT Id, Type 
      // FROM CollaborationGroupFeed 
     //  WHERE ParentId = :c.id];
     //     System.assertEquals(1, posts.size());
        }