You don’t have to convince them to come into the Catholic Church.
You simply have to convince them that to know God is to love Him, and if they want to love Him more fully they need to know him more fully - have them start with the church fathers and the early church. If they follow the history of the church with an open mind, they will see the truth of the matter.
I think the problem with Evangelical churches or Catholicism-Lite is that it’s “good enough” for them. It fits their needs because their needs are shallow.
Sounds bad, I know. But they really don’t seek heaven, they seek assurance that they are bound for heaven. How can anyone know that they are going to heaven when they don’t even know what’s going to happen tomorrow? All fellowship is is the surrounding oneself with like-mineded fellows and sharing feelings and experiences; if your common experience is superficial, can you really create depth? But I digress.
However, that is the obstacle that you have to overcome. Their hearts, God bless them, are in the right place. Currently in America, they are a more formidable force than Catholicism because even with their parital grasp of the Truth, their fellowship allows them to be that common voice; if only more Catholics were that way.
I think I may have stumbled upon the problem: they see the lack of fellowship/unity in catholic America. As a group, we vote no differently than the whole, and when all of the “cultural catholics” and self-proclaimed catholics, and cafeteria catholics weigh in, we just don’t look that convincing. I wonder what they might think if they attended a catholic conference where othodox catholic teaching was the rule…