OK, looks like WordPress discards everything from the twitter page source code except the tweet and related info it publishes, provided the twitter URL is on a line by itself. As you noted, somewhere in the process, Insert PHP interferes.
This is a temporary fix:
At line 58 of the plugin's insert_php.php file you'll see
add_filter( 'the_content', 'will_bontrager_insert_php', 9 );
change that to
add_filter( 'the_content', 'will_bontrager_insert_php', 7 );
(The only change is the digit 9 to 7).
Verify the Insert PHP isn't now interfering with any of your other plugins.
The digit can be brought still lower if needed for your current or future WP versions. But no lower than necessary to let it process the tweet page URLs. The digit is a priority number and the plugin shouldn't usurp any priority it doesn't actually need.