In general, sadness is something to be avoided by anyone trying to serve G-d, like is understood from Tehillim 100:2 which says:
Serve G-d with joy, come before Him with song.
Contrary to what you suggest, There is no "rebuke" being made by the Alter Rebbe. He only applies the comparison of being like a heretic, meaning similar to someone who denies G-d and the Torah, to one who has actually attained the level of comprehension he describes in the opening of the letter and who then indulges in excessive grieving over material hardship.
Even there, he only points out the hypocrisy of denying G-d's goodness for such a person by overindulging in being 'grief-stricken' over material hardship.
He explains this in the context of understanding that because part of our faith is that the universe is created by G-d, Ex Nihilo, meaning from nothingness, constantly, every moment of every day, and that we also believe that nothing negative comes from G-d, like is found in Bereshit Rabbah 51:3, then it is understood that even those things which appear to us as negative, are in truth, only good.
The second part of your question seems to be confusing the difference between mourning which is אבלות, and sadness or depression (עצבות ודאבה).
Mourning is a procedure. It has specific practices which are outlined in halacha about what one is to do or to refrain from doing. In fulfilling those procedures, one has mourned, fulfilled a mitzvah and served G-d.
Sadness and depression are emotional states. They are actual states of being, but are not mitzvot. There is no commandment of G-d to be sad.
The Alter Rebbe clearly acknowledges the obligations for mourning as can be seen, for example, in Siddur Torah Ohr, volume 2, section 6 and 7 discussing the laws of mourning. With the petirah of a loved one, most people experience sadness. But to be sad is not some requirement of mourning.
To my knowledge, the Alter Rebbe does not advocate suppressing emotions. What he teaches is that emotion, middot, arise via the intellect (Mochin or Sechel). That is why his school of Chassidic thought is called 'Chabad'. It refers to Wisdom, Understanding and Knowledge (חכמה בינה ודעת), the components of the intellect. It means that part of our Avodah is to strive to use our intellect, each individual according to their unique capacity, to express their emotions properly in service to G-d.
There are two stories which come to mind to help illustrate this idea.
The first is from the bottom of Makkot 24a and continued on 24b. It recounts two occasions when Rabban Gamliel, Rabbi Elazar ben Azarya, Rabbi Yehoshua, and Rabbi Akiva were walking together along the road. In each circumstance, Rabbi Akiva's response upon seeing a very negative event was laughter from joy, while the other sages cried from sadness.
The difference in emotional response was due to how each individual processed the observation intellectually. With Akiva, he saw in the negative events the revelation of G-d's goodness, meaning that only good comes from Above like is mentioned above from Bereshit Rabbah. With the other sages, that goodness was not perceived. It was concealed from them. As a consequence, they cried in sadness.
The second story recounts what happened when the 5th Lubavitcher Rebbe, Rabbi Shalom Dov Ber Schneerson was passing away in front of his son, the future 6th Rebbe, Rabbi Yosef Yitzchok Schneerson.
The 6th Rebbe was known throughout his youth as being a highly emotional individual. His father had worked with him throughout his life to master his mind, his intellectual powers, in order that his emotions should always express properly in service to G-d.
In these final hours of the 5th Rebbe's life as he lay before his son, he was giving his son final guidance and instructions about becoming Rebbe in the coming hours. In that moment, Rabbi Yosef Yitzchok was overcome with grief and sobbing. As incredible as it may seem, his father, Rabbi Shalom Dov Ber's response to his son was, "Middot?!...Middot?!...Mochin!"
With all this, it is important to remember that the Tanya is also called Sefer Shel Benoni, the Book of the Intermediate Person. That means that the level of service to G-d described therein, is attainable by anyone who stands in that middle ground between being totally righteous or the opposite. However, with that said, it also explains that this is a constant, day to day struggle for such people. That is why it is called Avodah, work.