I've tried the suggestions in this thread, and I still can't get the execute to test. Now, I'm having to work with someone else's code unfortunately and I have the impression it wasn't written correctly.

For example the query was actually built inside the start method, rather than outside the method as a class variable - this prevented any ability for me to use the LIMIT 200 in the test query.

I've now added the query as a class string variable, and am updating the query to include LIMIT 200 in the test before it enters the start method.

Also, the debug logs show the query is returning the test records I created. I'll try to simplify the code:

global class batchJob implements Database.Batchable<SObject>, Database.Stateful 
{
    global List<String> emailIdsList; 
    global List<Id> emailTargetIds;

    public BatchApex_PriceExceptionFieldUpdates()
    {
        emailIdsList = new list<String>();
        emailTargetIds = new List<Id>();
    }

    
    String str_SOQL = 'select email from Contact';

    global Database.QueryLocator start(Database.BatchableContext BC) 
    {
        return Database.getQueryLocator(str_SOQL);
    }

    global void execute(Database.BatchableContext BC, List<SObject> sObjects) 
    {
        list<Contact> Contacts = new list<Contact>();
        for (Sobject SO : sObjects)
        {
             Contact C = (Contact) SO;
             .... more code
        }
    }

    global void finish(Database.BatchableContext BC) 
    {
         .... more code
    }
}

Here's the test method:

static testMethod void invokeTestMethod()
{
    ... set up test records
    test.startTest();
    batchJob Batch = new batchJob();
    
    Batch.str_SOQL += 'LIMIT 200';
    Id batchprocessId = Database.executeBatch(Batch, 200);
}

Again, the query is returning records inserted by the testmethod, and the start method is working. The test however never runs the execute method.

So, given that the 200 record limit is being imposed on the batch query in the test, the query is returning records, and thre are no other errors in the debug log, what am I doing or not doing?

The job is more complex than this as far as the internal code for the execute and finish methods, but since neither of them are being called, this isn't impacting the result. However, if it will help I'll add the rest. Otherwise, the process seems to be working in practice, but I can't improve the test results until I can get the rest of the methods to run.