In case anyone else runs into this, here's how I solved the issue. In admin/components/com_mtree/admin.mtree.php I made the following changes...
I searched for this:
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 $img_id = $this->_db->insertid();
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It's around line 1359. After that line I added a new line:
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 //DB Get Current Time$DBtimestamp = time() . '_';
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Then I updated a few lines below to include the timestamp:
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 // DB Add timestamp to filenamerename( $old_small_image_path, $mtconf->getjconf('absolute_path') . $mtconf->get('relative_path_to_listing_small_image') . $DBtimestamp . $img_id . '.' . $file_extension);
 rename( $old_medium_image_path, $mtconf->getjconf('absolute_path') . $mtconf->get('relative_path_to_listing_medium_image') . $DBtimestamp . $img_id . '.' . $file_extension);
 rename( $old_original_image_path, $mtconf->getjconf('absolute_path') . $mtconf->get('relative_path_to_listing_original_image') . $DBtimestamp . $img_id . '.' . $file_extension);
 
 // DB Add timestamp to filename stored in database
 $database->setQuery('UPDATE #__mt_images SET filename = ' . $database->quote($DBtimestamp . $img_id . '.' . $file_extension) . ' WHERE img_id = ' . $database->quote($img_id));
 $database->query();
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I had to do the same thing for the copy listing function in admin.mtree.class.php (around line 1435.)
Chris