There are the general blessings of wearing the Brown Scapular :

Promises of protection in life and at the hour of death, and a special intercession for release from purgatory. 

And there are the Sabbatine privilege's that you can obtain wearing the Scapular if you follow the requirements:
Fasting from meat  on Weds and Saturdays
Reading the "Little Office of THE Blessed Virgin Mary' ( can be commuted to the Rosary by a priest)
Observing Chastity in  one's state in life.

I was given the understanding that your first brown scapular should be blessed, but the many subsequent ones that come after are under the same blessing and it does not have to be repeated.

I believe that's what I heard as well, that each subsequent one needn't be blessed, just the one used at time of enrollment.  I believe it's the case because I think the blessing is actually more of you the wearer of the scapular than of the scapular per se.  I'm sure it doesn't hurt though.  I do believe the medals have to be blessed each time though.  But we should confirm, since my memory is rusty.

I felt bad the one time at an SSPX chapel where I took a new rosary I had bought (one of the super-tough ones with the paracord, etc. ... since mine broke so often) and took it the rail since they did that after, say, the 2nd Sunday of each month (can't recall exactly).  Well, this particular Rosary also had some medals attached, and I saw the priest going through looking for the blessings for each medal, a St. Benedict's, St. Joseph, St. Jude, Miraculous, etc.  I think there are like 5 medals attached to this rosary ... which is rather heavy, made with metal beads and the paracord rope ... so that in a pinch it could easily double as a physical weapon, not just the spiritual one -- since this can do some damage if properly wielded.  But the priest felt the need to bless each item separately, where he'd look up the special blessing for each one, and I could see him paging through looking for them.

I actually highly appreciate that.  Many priests just make the sign of the cross and apply holy water ... which is technically OK, but the Church has a great treasury of blessings that are both beautiful and powerful, and I don't like the lazy approach there.  So I was grateful that he used the specific blessing ... though it was not my intention (as I didn't anticipate it) that he would bless each individual medal attached to the Rosary.

https://en.wikipedia.org/wiki/Fivefold_Scapular

It's possible that the Traditional Redemptorists might have the ability to enroll ... but not 100% sure, since I'm not sure what would or would not be supplied in that regard given the current Crisis.  Perhaps you could track down an elderly Redemptorist who's a valid priest still and be enrolled.

From the site you posted.

 "in 1886 Pope Leo XIII gave permission to bless and enroll the five scapulars cuмulatively, and later the Church extended the faculty (to bless and enroll the Fivefold Scapular) to any priest.

Prior to the 1950's priests had to apply to Rome for faculties to impart specific blessings/indulgences to certain sacramentals.  
As an example it was the custom for people to seek priests of a certain religious order or those preaching a mission who held special privileges to impart certain indulgences to a rosary.  They would have them bless their rosary increasing the indulgences to be gained when it was prayed.  Bishops usually had the faculty to impart all indulgences.

Sometime in the early 1950's Pope Pius XII closed the department in Rome where priests had to apply for these special faculties and extended the privilege to ALL PRIESTS.

I actually remember this happening, as the nun's told us about it in school, it was announced from the pulpit and was printed in the church bulletin.  The four priests at our parish must have been kept really busy for a while after as everyone would have wanted the indulgences increased on their rosary.
I also remember when after being confirmed the Bishop blessed all the rosaries the class had received for their confirmation with the A B C D indulgences (Apostolic on the Crucifix, Benedictine, Crossier and Dominican)

Oh, for those looking, I ordered from here the last time I needed a new 5-fold scapular, and this was by far the highest quality 5-fold scapular I've found yet ... I highly recommend it.  Nice attention to detail along with the picture panel put on each one individually.  Most of them did not have that for the brown, blue, and black sandwiched in between the white and red (but just on the white and red), where they just slap it together.  For this one here, each one could actually serve as a quality individual scapular if it were separated from the others.

https://traditionalcatholicstore.com/shop-our-store/five-fold-genuine-wool-scapular/

Lad, have you noticed that they have switched the black scapular of Our Lady of Seven Sorrows (Servite) to the Passionist black scapular (different devotion, actually a duplicate of the red Passion scapular) ?

Both the Ritual Romanum 1947 Pg 524 and the Parish Ritual 1962 Pg 281 specifically mention receiving the Servite habit in the Blessing and Reception of the Five Scapulars. 
It's pretty obvious the Five Fold approved by the church belong to :
"The habit of the Order of the Most Holy Trinity
The scapular of the Passion of our Lord Jesus Christ
The scapular of those dedicated to the most blessed Virgin Mary
The habit of the Servites
The habit of the Society and Confraternity of the Blessed Virgin Mary of Mount Carmel"
Quoted in part from the blessing.

Some other five Folds contain St. Joseph and one St. Michael so I don't know how the priest could change the blessing of the church to accommodate them, if he even noticed they were included.
It appears that in their zeal for a specific devotion they've altered what the church approved.
Be careful what you buy