The arms themselves looks pretty genuine to me. There are a couple of ways that you can check this out...
1. place a request for pedigree information with the College of Arms in London for your G-G-Grandfather - this should be addressed to The Officer in Waiting via the College of Arms web-page at ...http://www.college-of-arms.gov.uk/contact-us
2. have a fully documented paternal pedigree research done going back from your G-G-Grandfather which should give you some info on the paternal bloodline that might link up with one of the entries shown on the attached scans of Fletcher entries in some books that I have.
The other source information that you mentioned in your post would be from one of the Visitations - i.e., Leicestershire 1616 or Warwickshire 1619, or 1682-3. Do you have any domicile information for your G-G-Grandfather as that might be of assistance.