Thank you for the additional information. I am sorry to hear of your dilemma. I realize how frustrating this is for you and I hope to provide you information which is accurate and useful, even though it may not be the news you were hoping to get.
The dealer would not, as a general rule be liable for the obligations of the third party warrantor. The warranty would have a process in its written terms for disputing a denial of claim. You need to find that provision and initiate that appeal. You may have to go to arbitration before you can sue.
It is important that you follow the procedure in the warranty to the letter, as a failure to do so could prejudice your rights to appeal.
Toy should file a consumer claim, as well, with the Florida Attorney General's Office.
You may want to retain local counsel to write a threatening letter to the warrantor and see if you can scare them into covering all or some of the repair.
If you need assistance finding local counsel try Martindale Hubble (site rules prohibit us from referring specific attorneys) . Many attorneys themselves use this site to locate attorneys outside their jurisdiction or expertise:
It is a huge worldwide database searchable by location and specialty. The attorneys are all peer rated. So, they represent the top of the profession.