[SalesForce] Post into chatter group without access to the group

When a field is populated on an Opportunity record I want to alert a team of users of that. I don't want to spam them with emails so I thought to make a chatter group and then post chatter feed items with the details that they can review whenever.

I wrote this method and it seems to work posting the link to the Opportunity:

public static void competitionAlert(Map<Id, Opportunity> newMap, Map<Id, Opportunity> oldMap)
{
    // Find the Chatter group Id
    CollaborationGroup chatterGroup = [ SELECT Id, Name 
                                        FROM CollaborationGroup 
                                        WHERE Name = 'Competitive Intelligence'];

    List<FeedItem> feedPost = new List<FeedItem>();

    // loop the Opportunities to see if the compeitition field changed.
    for (Opportunity  oppy : newMap.values())
    {
        if(
            oppy.Competition__c != oldMap.get(oppy.Id).Competition__c &&
            oppy.Competition__c != null
        )
        {
            // create a chatter feed post alerting the team
            FeedItem fpost = new FeedItem();
                fpost.ParentId = chatterGroup.Id;
                fpost.LinkURL = URL.getSalesforceBaseUrl().toExternalForm() + '/' +oppy.Id;
                fpost.Title = oppy.Name;
                fpost.Body = 'An Opportunity was flagged as "Closed Lost" to ' + oppy.Competition__c + ' please review.';
            feedPost.add(fpost);
        }
    }
insert feedPost;
}   

The challenge is that the chatter group needs to be private, so when I lock the group down I get errors when a user who does not have access triggers the method and tries to create the post.

Is there a way to post into groups without having access to the group or a way to post as a user who does?

FINAL Solution:

Handler Class:

public static List<Opportunity> competitionAlertFilter(Map<Id, Opportunity> newMap, Map<Id, Opportunity> oldMap)
{
    List<Opportunity> oppy = new List<Opportunity>();

    // loop the Opportunities to see if the compeitition field changed.
    for (Opportunity  o : newMap.values())
    {
        if(
            o.Competition__c != oldMap.get(o.Id).Competition__c &&
            o.Competition__c != null
        )
            {
                oppy.add(o);
            }
    }
    return oppy;
}

public static void competitionAlert(List<Opportunity> chatFilter)
{
    ChatterPosts.createCompetitiveIntelligencePost(chatFilter);
}
}

Extender Class

public without sharing class ChatterPosts {

public static void createCompetitiveIntelligencePost(List<Opportunity> chatFilter)
{       
    // Find the Chatter group Id
    CollaborationGroup chatterGroup = [ SELECT Id, Name 
                                        FROM CollaborationGroup 
                                        WHERE Name = 'Competitive Intelligence'];

    List<FeedItem> feedPost = new List<FeedItem>();

    // loop the Opportunities to see if the compeitition field changed.
    for (Opportunity  oppy : chatfilter)
    {
        // create a chatter feed post alerting the team
        FeedItem fpost = new FeedItem();
            fpost.ParentId = chatterGroup.Id;
            fpost.LinkURL = URL.getSalesforceBaseUrl().toExternalForm() + '/' +oppy.Id;
            fpost.Title = oppy.Name;
            fpost.Body = 'An Opportunity was flagged as "Closed Lost" to ' + oppy.Competition__c + ' please review.';
        feedPost.add(fpost);
    }
insert feedPost;
}    

}

I'm excited about this methodology because for future use cases I can pass whatever data to the ChatterPosts class and not run into this issue again.