In the advanced options do you have it set to log API errors? With that enabled, if the response from the Google API contains an error, it will create a log file inside of the plugins ‘log’ folder on the server. If logs are enabled, and that folder is blank, that means the response from Google is good.
I understand not being able to give out access to the site, I usually start there just because it can be very difficult to diagnose these issues without being able to take a look at things directly.
With that said, what I normally do when given that access, is to edit the plugins core.php file to add in some additional error logging to see what is happening.
The following code snippet can be added into some functions to gain some insight into what is going on:
file_put_contents(GPI_DIRECTORY . '/logs/DEBUG_LOG_X' . '.txt', print_r($THE_THING_TO_LOG, true), FILE_APPEND);
Change the letter X in the filename with a unique number for each item you want to log, and change $THE_THING_TO_LOG to a variable from the function you want to log. Keep in mind that these log files append, so new data is appended to the bottom of the file whenever data is written. You will want to make sure to remove any logging you add when you are done so that the files do not keep filling up with data.
I will often use this right after line #180 of core.php to log the $result variable, which contains the response from google.
Also, Changing line #284 to:
$db_response = $wpdb->insert( $gpi_page_stats, $gpi_page_stats_values);
And then logging the value of the $db_response variable right after it would give you the response from WordPress when trying to store the data. If the new log file is created on the server, but it is blank, then the operation failed, and this would suggest that there is something wrong with the tables or the permissions. If the log file has a 1, that means the save operation was successful. And if the log file isn’t there at all, that means things failed before it ever got to this step.
Another thing you could do, is ensure wp_debug is turned on inside of your wp-config.php file, and watch for any errrors. If you have access to your PHP error logs, you can check there for any fatal errors or warnings.
Let me know what kind of data you see while running any of the above diagnostics and I will do my best to help interpret. If you are able to provide me access to the environment, or you are able to reproduce the issue in a separate environment that I am able to access, I would be more than happy to help you look.