What you say is openly contradictory. They receive certain exemptions due to their size, but their tax bill has nothing to do with their size. ???
Can we please stop arguing like influencing is only true if it is done in the most direct way (similar to the quid pro quo debate). Obviously if big company lobbyists try to get tax law in their favour they are not pushing for "please write a law which exempts companies over N employees from taxes." They push for laws that sound innocent but only they will be able to take advantage of, just like it is at the moment. The outcome is still the same they pay less or zero taxes.