In this sentence big-time is used as a verb:
In short, polite people never big-time you; instead, they always make you feel big time.
What does big-time mean?
As a (slang, informal) verb, to "big-time" would mean to act in a manner that projects the idea that one is the more important person or entity in the conversation or relationship or business deal. As an adjective, "big-time" would mean "important, significant, worthy".
The company does not big-time its customers. It makes the customer feel big-time.
An old-fashioned locution that means much the same thing is "to lord it over".
The company does not lord it over the customer. They make the customer feel like he's the king.