What does this dialogue mean in The Barefoot Contessa (1954)?

Harry's commentary on Mr. Kirk Edwards' money making him more attractive is indeed worded quite interestingly and might seem a bit contradictory at first. It is not entirely clear if Harry really believes money makes men more attractive or not, but that's not what the line is about. We need to concentrate on the phrase "feel" there, as Maria's assumption isn't about Harry's or Mr. Kirk Edwards' money and its influence on being loved, rather than about the men's attitude towards that.

When he says that the rich men he knows feel loved "in spite" of their money, he means that those rich men think that being loved doesn't have anything to do with their money, no matter if that's true or not. Ergo they don't think their money makes them more attractive to women, which does fit to the sentence before.

Of course this is ultimately a cynical, if not spiteful, commentary and it's not clear if Harry really believes that. What he sheds a light on here is the fact that every rich man thinks he is loved not for his money but for himself, if that is true or not isn't quite the point. In fact I'd say Harry doesn't think it's generally true rather than that many rich men are blinded by their money. The contradiction that they are, directly or indirectly, using their money to impress women, yet tell themselves they are loved for their own. And he does that by taking Maria's suggestion of him being intimidated by money and turning it around. It is about what effect men think (or tell themselves) their money has on women.

We might even go as far as saying that he is not intimidated by money because he knows that the above statement about rich people feeling loved in spite of their money doesn't reflect the truth rather than being an illusion those rich men fall to and that they are mostly loved for their money, yet believe they aren't. And that is something Harry doesn't want for himself.

In fact this goes quite well together with the other issue about him not meeting a lot of "women". As you already guessed yourself and as Yasskier explains in his answer, he makes a distinction here between women and girls, particularly based on their emotional maturity. And he foots this distinction on the same ideas of superficiality that the commentary about rich men being loved for or in spite of their money cuts down to.

It is not entirely clear if he thinks that all "actresses, female writers, singers, painters, [...] female agent" aren't "women" or not, but it's clear he thinks rarely anyone of the ones he met is. It might be relevant that those are all professions from the arts, even more so professions that might come with a certain success and societal esteem, but supposedly also vanity and superficiality. What he is saying is that in his eyes women from these trades aren't emotionally mature enough to count as "women" for him. Yasskier explains this quite well in his answer.

And of course the fact that the "woman" he dates is a "script girl" adds some ironic sugar on top of that and that is very well placed there. The term "girl" in her title reflects on the respect (or lack thereof) society has for her profession. She is a "girl" in the eyes of society, which doesn't quite value her profession as much as that of an actress or a female writer (note that it isn't "acting girl" or "writing girl"). Yet she is a "woman" in the eyes of Harry because her supposedly lower profession frees her from the superficialities of the "high society".

And this relates back to the above statement on rich people. It is the "girls" that their money attracts, not the "women". The whole dialogue figures into a broader commentary on the superficiality of society, where people are attracted by money, use money to impress eachother, yet think of themselves a lot higher and believe their money doesn't have anything to do with their success, because they lack the maturity of grown men and women.

1. Rich man issue:

What makes women think that having money makes a man feel more attractive? I never met a rich man yet who didn't think he was being loved in spite of his money.Or at least, this is what they think.

It means that Harry doesn't believe that having money makes someone more attractive, as every rich man that he knows is being loved for other values than his money (and Harry follows here the believes of his rich friends). So if those rich men became suddenly poor (or never had money in the first place), they would be still be loved.

Still confused? Try replacing the word "money" with, let's say "dog". Harry would be saying that no man that he knows who owns a dog is being loved ONLY because he has a dog.

2. Woman vs girl

Have you ever heard a saying "This [insert some 'heroic' action] will separate men from boys" (eventually "turn boys into men)? Well, Harry says that he can find a difference between "Girl" and "Woman".

A girl is someone with whom you can have some fun. She might be attractive, she might be smart, but the mutual attraction won't last, because it is superficial. A woman, on the other hand, is someone who you can relate to, someone who is mature that can look at you past the initial impression. It is someone, who could stay with you in good and bad through your life.

Actresses (according to Harry) belong to the first category - while they are pretty and famous, their love is fickle, as their careers depend on popularity, so they crave constant attention from the masses. A woman with a more mundane job, like a script girl, would be able to develop a deeper, more meaningful relationship, since she doesn't need to convince the whole world to love her all the time.