Your complaint needs to go to Royal Sun Alliance who are a very reputable company in the insurance business. Lloyds have only put their name against it and brokered the insurance as so many other large companies do as it gets their brand onto something that they can sell as their own product under another large (product specific) company and this allows both companies to enter into markets and sales that they might not ordinarily get access too.
If RSA have declined the claim then they will have their own reasons which Lloyds will not be party to. If you feel that the claim has been declined incorrectly and the process dealt with badly by the insurrer then the appropriate complaint needs to go to RSA directly as per their complaints process and they will need to resolve or dismiss the complaint themselves. They have up to 8 weeks to do this and only then can you move things forward to the ombudsman. Unless you have fully exhausted the complaints process then the ombudsman will only redirect you to do this first and you start at the beginning of the 8 week process again.
While I fully see your point that it was Lloyds that sold you the product they will argue that while they promote and "broker" the product under their brand it is actually underwritten and controlled by the insurer. To give a realistic comparison to this - I sell home insurance and all other types of insurance to my clients under my company name, I broker and promote their products and I will alway send out my own branded letters etc (the letter will always say who the insurer is though) but the policy will always belong to and be the responsibility of the issuing and underwriting company....although if any of my clients did call me regarding an issue with any provider (and they have done several times) I would always liaise with them to resolve the issue one way or another.
Get back in contact with Royal Sun Alliance and liaise with them as to how you can resolve the claim, if you then feel you need to complain then do so to RSA and not to Lloyds.
Good Luck