I am a fulltime Certified FileMaker Developer. Want I understand you want me to do is write several scripts to log-on to the Google API and receive a token, which is required before you can use any of the other Google API functions.
Once I have a good token, I will have another script which will gather data from several FileMaker fields for To, Subject, Body and will assemble that data into JSON which will be passed to the Google Mail API to send your email.
As a 'cleanup' step, I will add a test condition to the second script, so if you attempt to send an email without a valid session token (which expire), it will automatically re-login into the Google API.