For the exception , I believe you are initializing error = '' and errorRecCount in wrong place. It should be inside the for loop like this:

for (order ord : upsertOrderList) {
    error = '';
    errorRecCount = 0;

As explained here order of the result and the upsert records will be always same.

for (Integer index = 0; index < lsr.size(); index++) {
    error = '';
    errorRecCount = 0;
    if(lsr[index].isSuccess() && lsr[index].isCreated()){
        if (lsr[index].isCreated()) {
            successRecCount++;
        }
        orderIdSet.add(lsr[index].getId());
    } else {
        hasErrors = true;
        for (Database.Error err : lsr[index].getErrors()) {
            errorRecCount++;
            Order_Integration__c ordInt = new Order_Integration__c();
            ordInt.Id = upsertOrderList[index].Order_Integration_Ids__c;
            error += err.getStatusCode() + ': ' + err.getMessage();
            error += 'Fields: ' + err.getFields();
            error += '\n';
            ordInt.Status_Description__c = error;
            ordIntegrationList.add(ordInt);
            if (err.getStatusCode() == system.StatusCode.REQUIRED_FIELD_MISSING) {
                if (errorMap.containsKey(err.getMessage())) {
                    count = errorMap.get(err.getMessage());
                    errorMap.put(err.getMessage(), ++count);
                } else {
                    errorMap.put(err.getMessage(), 1);
                }
            }
        }
    }
}

For the exception , I believe you are initializing error = '' and errorRecCount in wrong place. It should be inside the for loop like shown above.

First your core logic of checking if a record is success or not is incorrect. . As it will always create junk Order_Integration__c records for each upsertOrderList.

As per your code lets say if first result in lsr is not successful, so the code will go ahead iterate on all upsertOrderList and create records for all these in else condition.

For checking if a upsert has passed successfully or not check this

For the exception , I believe you are initializing error = '' and errorRecCount in wrong place. It should be inside the for loop like this:

for (order ord : upsertOrderList) {
    error = '';
    errorRecCount = 0;

The issue is as you have initialized String at top and concatenating it always, so there might be scenario in your batch where that String is growing beyond 12MB of heap limit.

First your core logic of checking if a record is success or not is incorrect. . As it will always create junk Order_Integration__c records for each upsertOrderList.

As per your code lets say if first result in lsr is not successful, so the code will go ahead iterate on all upsertOrderList and create Order_Integration__c records for all upsertOrderList in else condition.

For checking if a upsert has passed successfully or not check this

For the exception , I believe you are initializing error = '' and errorRecCount in wrong place. It should be inside the for loop like this:

for (order ord : upsertOrderList) {
    error = '';
    errorRecCount = 0;

The issue is as you have initialized String at top and concatenating it always, so there might be scenario in your batch where that String is growing beyond 12MB of heap limit.