Catechism classes?  I wonder if the classes are for the parents rather than the child...who hasn't even reached the age of reason yet.  Clearly, people in that parish aren't regularly baptizing their newborns.  Before escaping the Novus Ordo we were at one parish that did baptisms only on the last Sunday of the Month.  We were there with our 3 week old daughter and there were several other children being baptized.  It was a long time ago so I don't remember how many, but it had to be close to ten more baptisms.  Every single child other than ours was old enough to walk.

I'm guessing--but it's only a guess with the Novus Ordo--that if a child is unbaptized and older than 6, the child would learn the faith in the regular catechism classes for his/her grade level before being baptized.  

In any event, a $100 fee for catechism in order to be baptized sounds rather steep to me.

As the OP does not provide the name of the parish nor a link to an online bulletin for more context a reply is necessarily speculative.
 
From a relative’s house I once bought I inherited a lot of parish bulletins from the 1940’s and 1950’s (she traveled a lot).  I wish I still had some for reference, but they got recycled before a move, I needed to downsize.  In large urban parishes with multiple priests' baptisms would be most Sundays or twice a month (as announced in the bulletin) in the afternoon.  The newest junior assistant pastor would get sent over to handle it.  Families would just show up with the infant and godparents and fill out a form so the certificate could be mailed to them (I have my certificate and the envelope it was mailed in), though some parishes may have had a preregistration process.  The names would be printed in the next Sunday’s bulletin.  In smaller parishes a baptism was often just before or after Sunday Mass and would be mentioned in the announcements.  As a child I recall a couple of these where one or both of my parents were the godparents.  A stole fee was often involved, I think.
 
Coming up to current times many parish priests know that too often a request for baptism may be for cultural reasons or so “granny doesn’t disinherit us”.  That has always been an issue, but more so now.  It seems prudent to require parents and godparents to attend a class in these times, to be sure that they know the meaning of Baptism and their obligations as parents and sponsors.  A fee is often asked, which would cover the cost of any materials distributed and the stole fee (similar to a Mass stipend).  Some families may have never passed their hand over the parish collection basket, those who are regular contributors can adjust their weekly offering if they feel it justified.  Most pastors would grant a dispensation for legitimate circuмstances.
 
As for “age six and under”, a child over age 6 or 7 is considered to have reached the age of reason (per Pope St. Pius X) and ought to receive catechetical preparation before Baptism.  Trent speaks to the distinction between infant and older candidates for baptism, but I don’t have time to look it up now.  The child would probably be put into a First Communion class, with some additional instruction.


I recall a priest who said that baptismal and marriage preparation were two of the church's most fertile opportunities to reach the fallen away.

We are CMRI.  When I was a catechist teacher for New Order in the early 90's, I had students ages 8 to 13.  Not baptized.  Boy did my head ponder, what is going on?!  I was given a copy, one sheet of paper explaining the Gospel for that Sunday.  That was it!! There was no teaching, preparation for baptism!  I snuck in copies of prayers for the rosary.  The secular nun told me it was too soon for the students to learn their prayers!!!  Then I decided to have a movie, The Song of Bernadette. We got half way through and left the rest for the following Sunday. 

That Sunday, it poured down rain, big time.  I thought I just might have to stay home.  I drove it, and parents came in with their children, the students.  They wanted to see the movie!  The secular nun told me that another teacher was going to have a movie that day, HM?  I said, "Well, tell these parents who are here to watch the movie, no they can't.  They drove in the pouring rain to see the finished movie. So, I got my way!

Now, if you are in the new order, get out!  The attitude and behavior (actions) tell you that their Knowledge of the Beauty and power of the sacraments, means money to them!  Souls are not in their minds.

In the 60's, there was baptism of babies every Sunday.  If it was a wait of longer than 4 weeks, it could be mortal sin.  Parents don't wait!!

We had our son baptized four weeks after he was born, I wanted it sooner, but couldn't make that happen.

One reason we opted for a Catholic hospital was that we thought the personnel would know what to do if, God forbid, he was in extremis.  When we checked in, I soon got the vibe that they had absolutely no clue as to such a thing.  I ended up taking a bottle of holy water and leaving it on the shelf as long as we were there, and kept one in close proximity until the day of his baptism.  (They had these nice little suites, more like a studio apartment, and I slept on a very comfortable fold-out bed during our stay.)

In my day, 40’s, Catholic women went to St Margaret’s for their births, and Sister Anne insisted that mother needed 10 days for R&R. There was always the offer to have Baby baptised during that 10 days. She was at or nearby for practically every birth.

In the 40's, Mother worked but in the home. Father brought home the bacon. They managed on one paypacket. Yes they were different times