Different Webhost and using Google Mail server: Is there a better way than SMTP?

Currently my situation is the following:

I am using Wordpress which handles a few things mail related: Registration forms that send a notification and contact form that send a message. However, because the mail server we use it a remote mail server connected to with SMTP, the contact form is confusing. The email that is received by the contact form should say it is from the person who fills the contact form out, but is instead forced to conform to the SMTP email settings and says it is from the SMTP email.

We want to use Google Apps Business and their mail servers, but according to the Gmail API (https://developers.google.com/gmail/api/ and https://cloud.google.com/appengine/docs/php/mail/#Sending_Mail) to connect to these mail servers, we'd have to use SMTP or IMAP.

We know we can create a dummy email for SMTP settings and have the website send using that configuartion, but we need another solution.

Is there a way into making Wordpress think that the Google mail servers are local mail servers so that phpmail can be used properly?