"none of the other regiments organized during Hindman's tenure were referred to as Trans-Mississippi regiments, only those Arkansas infantry regiments assigned to Price's Division"

But please note that:

SPECIAL ORDERS,No. 39.
HDQRS. TRANS-MISSISSIPPI DEPARTMENT, Little Rock, Ark., September 28, 1862.
I. 1st. McRae's brigade, consisting of McRae's, Matlock's, Johnson's, Pleasants', McNeill's, and Morgan's regiments of Arkansas Infantry, and Woodruff's Arkansas Battery, will move on Monday next, 29th instant, and report to Brig. Gen. J. S. Rains, at Elkhorn. [ar19_883]

The six named regiments are exactly the six TM ones [2nd, 4th, 6th, 1st, 5th, 3rd respectively]. It is difficult to call it a curious coincidence. I refuse to believe it.

The interesting question should be, why the 24th, 27th [#1] and 33rd Ark had not a "special" name, like their "Trans-Mississippi" and "Northwest Division" brethren. The date of organization looks more or less the same.

[#1] On Wikipedia there is a reference to the 27th as "1st Regiment, McBride's Brigade" but I believe this was standard bureaucratic jargon, meaning "the first regiment of one's brigade, listed in seniority order" (useful when you have to quickly find out who is the next in command). BTW the formulation of McRae's brigade above did not follow seniority order, unless an inordinate amount of colonels were absent at that time. Actually it did not follow any logical order I could think of.