By far the most effective strategy in apologetics is to meet your opponent on common ground. There are certain things on which you and your opponent are never going to agree naturally. It doesn't matter how much you argue from those points, your opponent is always going to reject those arguments. You cannot use the Bible to disagree with an atheist, for an example, because no matter how much the Bible affirms that there is a God, and those who don't believe in Him will perish, the atheist does not believe the Bible is true, and so he will not affirm its claims, and you will never get him to agree with you. However, most atheists believe in science, logic, history, and a range of other academic fields. Thus, we can take a claim from the Bible and show how one of these academic fields supports it. Once the atheist rejects the evidence provided, it is clear that their allegiance is to atheism only, and that not even science etc. are really their motivations.
In the same way, it is silly for Catholics and "Protestants" to engage each other using their claims to authority/lack thereof. When a Catholic asserts the authority of the Catholic Church, it will have no effect on a "Protestant" who rejects the Catholic Church's claim to authority. Stamp your feet, puff out your chest, bellow at full volume, a "Protestant" will not even blink. The only people who will be convinced by any allegedly authoritative statement from the Catholic Church are those who already accept that the Catholic Church has the authority to make those claims.
Now, the problem with the term "Protestant" is that it is far too broad a term. It's why I keep using it in air quotes, and why I refuse to label myself a Protestant. A Protestant is just a nominal Christian who rejects the authority of the Catholic Church. Because it is such a broad term, it can encompass a wide range of weird and wonderful denominations, including ones with their own dubious claims to authority. Mormonism, for example, holds three so-called scriptures as being authoritative: The Book of Mormon (which Joseph Smith called "the most correct book" and "the cornerstone of our religion"), Doctrine and Covenants, and the Pearl of Great Price. Of course, as a so-called "Protestant" myself, I utterly reject these documents. What's really telling here is that there is no name for a nominal Christian who rejects the authority of the Mormon Church.
Therefore, it is unfair, and frankly rather silly, to lump all non-Catholics together under the umbrella term "Protestant". Far better to describe people by what they do believe, not what they don't. Let the Mormons be Mormons, let the Catholics be Catholics, and let the Christians be Christians (Acts 11:26).
Now, what is the common ground between Christians and Catholics? The Bible. While not everyone the Catholic Church identifies as "Protestants" hold to Sola Scriptura (Scripture Alone), it is a recognisably non-Catholic, distinctly Christian doctrine. It is the belief that all Scripture is given by inspiration of God, and is profitable for doctrine, for reproof, for correction, for instruction in righteousness, that the man of God may be complete, thoroughly equipped for every good work (see 2 Timothy 3:16-17). Compare this with "both Sacred Tradition and Sacred Scripture are to be accepted and venerated with the same sense of loyalty and reverence", as we find stated in the Dogmatic Constitution on Divine Revelation, Dei Verbum, and we have our common ground. "Protestants" believe scripture is authoritative, with no equal. Catholics believe scripture is authoritative, with an equal. The alleged equal is fruitless for discussion, as "Protestants" do not accept what Catholics call "sacred tradition", but both sides affirm both the authority, and even divine origin of scripture.
Because scripture is common ground for both Christians and Catholics, we can begin asking the question as to which view is better supported by scripture? Keeping in mind that both sides also happen to believe it is inspired by God Himself, meaning if scripture supports a group, they are the correct group, no further questions. Yet, you will notice that Catholics are extremely hesitant to discuss the scriptures. In fact, in my experience, convincing a Catholic to stick to the scriptures is like convincing a cat to stay in a full bathtub. You may have the odd few that enjoy the water, but most of the time, you're just going to get scratched. Even if your entire case is "this is what the Bible says", you are all but guaranteed to hear "ah, but this Church Father says this, and here's what this Council said, and do you really think Martin Luther was smarter than all these bishops, and by the way, you wouldn't know which books belong in the Bible without us, and..." so on and so forth. Convincing a Catholic to stick to the scriptures, in my experience, is one of the most agonisingly difficult tasks I've ever had to do.
So, why do Catholics seem to be basically allergic to our common ground? Simple: The Bible is anything but common ground. For Christians, the Bible is literally the only ground (and before any Catholics start throwing in objections about Christ, or Peter, or the Church etc., I want to make it absolutely clear that I am using a separate metaphor here, and I am quite familiar with the verses you will use to disagree with me). It is like a desert, and we are camels. We are perfectly at home here. Then you have the arctic of Catholic authority over there, and Catholics are walruses. A walrus may be able to move in the desert, but it will not be comfortable. It's just not their home. In the same way, while a Catholic may be able to talk about the scriptures for some time, it won't be too long before the heat starts to get to them, and they soon have to flee to more comfortable ground. When a Catholic is caught misusing scripture, it won't be too long before they fall back on some extra-biblical source, or even just says "well, that's just your interpretation, but we have the authority to say your interpretation is wrong".
The aversion Catholics have to sticking to the scriptures is a direct result of their theological conflict with it. Like all heretics, it is possible for them to twist a few scriptures in support of their view, but it doesn't often take much more than reading a few verses on either side to show that their interpretation is not feasible. By contrast, Christians do not need to twist the scriptures to fit our views, because our view is simply that the Bible is true. If the Bible says there is one mediator between man and God, we believe there is one mediator between man and God. If the Bible says we are saved by grace through faith, not of works, we believe we are saved by grace through faith, not of works. If the Bible says all Scripture is given by inspiration of God, and is profitable for doctrine, for reproof, for correction, for instruction in righteousness, that the man of God may be complete, thoroughly equipped for every good work, we believe all Scripture is given by inspiration of God, and is profitable for doctrine, for reproof, for correction, for instruction in righteousness, that the man of God may be complete, thoroughly equipped for every good work. And we are free to believe the Bible exactly as it is written. We do not have to twist it to support our views.
Using our alleged common ground, namely the scriptures, it is clear beyond dispute which faith is correct. The Catholic Church, giving mere lip service to the scriptures, is a less legitimate sect of Christianity than the Pharisees were a legitimate sect of the Jews. Any Christian within the Catholic Church should immediately repent and leave it. Any Christian wondering whether to join the Catholic Church should immediately lay aside such a thought. Eventually, God will deal with Catholicism. Don't get caught in His crosshairs when He does.