For the Latin LOTH, the Neo-Vulgate is used.
The current structure of the LOTH allows for the use of mid-day prayer at any one of the three canonical hours. That one is required. For the other canonical hours, the Gradual psalms from the complementary psalter can be used, but those hours are not required for those not bound to choir.
What can be done is the mid-day prayer moved from day to day or week to week, from Terce to Sext to None. The General Instructions allow substitutions of Sunday psalms for the psalms for another Sunday but not, AFAIK, other psalms. Your plan would slice out a good part of the psalter unfortunately, including some very good psalms. Moreover if you use Ps. 4, 90 and 133 (as allowed by the rubrics) for Compline every day, you’ll miss out completely on many of the Compline psalms as they are repeated at the mid-day office.
At solemnities, the complementary psalter is always used, so solemnities follow the Monastic tradition.
A note though on the minor hours. In the Benedictine psalter, the complementary psalms aren’t used on Sunday and Monday. Instead the divisions of Ps. 118 are used through the minor hours (with or without Prime, the division is different if not saying Prime). It’s considered one of the most important psalms in the Benedictine psalter; verse 116 is the verse recited by every monk at his profession (and by every oblate at his or her oblation).
So following your idea, unless you were to do as monastics do and recite Ps. 118 at the minor hours of Sunday and Monday, you’d miss out on the major part of this psalm (including verse 116) altogether because it’s split over 4 weeks at the mid-day hour. A small tidbit, every time that verse is recited, it’s like a mini-renewal of profession for the monks and they always bow at the recitation of verse 116.
My own opinion but I like to keep what’s monastic and Roman separate and intact except for some minor monastic additions like the Kyrie before the Pater, and the invocations for the dead and absent brothers at the end. Technically the latter two are “extra liturgical” but since they are after the final blessing it doesn’t affect the validity of the liturgy. I don’t use the invocation for the dead after Vespers as the last general intercession in the LOTH is always for the dead so it would be repeating one’s self to do so.
The best source of Neo-Vulgate psalms for the LOTH is the Latin LOTH itself, or the Psalterium Monasticum from Solesmes. The latter has the advantage of including the cursing verses and psalms.
I have no idea about the use of the Grail psalter as I do the LOTH in French and Latin.