You should be more concerned about the thought of a 4 week psalter than the omission of Psalms.
While in my limited scholarship of liturgical history I am leaning toward the opinion that Abp. Annibale Bugnini perhaps made a bigger mess of the Breviary than he did of the Mass, there is something to be said for a four week cycle, at least as an option (and there is some prescedent for it in Church history). I find that a better arrangement than the use of a diurnal (popular with several here), which compleatly eliminates ALL
the Matins psalms (34 in total according to the Pius X schema; even more according to the Monastic and Tridentine schemas). At least with the four week cycle only three psalms are missing. As a footnote: there are I believe a few psalms that are only said during Advent and Lent, and one would have to intentionally include the Gradual Psalms in Daytime Prayer to have those.
Least I'm misunderstood, I am in no way disparaging those who faithfuly recite the Little Office of the Blessed Virgin Mary (and God bless them for that), yet that only has 28 psalms in its entirety, leaving 122 NEVER recited. Ironically one would recite more of the Psalter by following the weekday and Sunday Novus Ordo
Lectionary (which restored the use of full psalms to the Mass).
Just my thought
I appreciate the delima of the OP; I will say that I started out with Christian Prayer
(the LOTH with an abridged Matins or Office of Readings) and found it a source of spiritual sucor, and in spite of its limitations and "defects", preferable to not reciting any office at all. If this is a particulary important discernment on your part you perhaps should search for a traditional community, despite the distance involved. I would assume that part of the monastic day is given to private prayer and spiritual prayer in the cell? Could not the traditional office be recited then?
Aren't you conflating those traditionally bound to the full office with laypeople and those in third orders and apostolic orders?
For the former, saying all the Psalms each week seems to be traditionally very important. But for the latter group, a diurnal or the Little Office is wonderful and manageable.
If a Benedictine monk is praying full office with all Psalms each week, he's doing his basic prayer. If a layman adds the Little Office to his daily rosary, or picks up the dirunal, he is probably (knowing nothing about the spiritual state of this particular layman) doing a wonderful extra work that takes serious committment especially if he has a job or is a student, has a family, other obligations, and so on.
To bring it back on topic: if Laetare joins a true monastic order, it would make sense for him to avoid ones that use the Little Office or the dirunal only as well as the LOTH.
But laypeople here on fisheaters do nothing wrong, and do something good, when they use the diurnal or the Little Office. Apples and Oranges.