I would rather say restricted moons than restricted months
because it is not the moons but the months that make up the YEAR.
The moons make up series of the moons (عِدَّةَ الشُّهُورِ as it is called in 9:36),
which is 12 moons or 13 moons. (This year it is 13).
The time of the 4 restricted moons
is the time when it is resricted to hunt wild animals (5:94, 95)
because they give birth to their babies and tend to them.
That is another proof that the 4 restricted moons are consecutive.
You may be on to something:
Shiin-ha-Ra = make it notable/known/manifest/public/infamous (in a bad or good sense), hire/contract for a month, stay a month, month old, month,
new moon, moon, full moon, any evil thing that exposes its author to disgrace, a vice/fault or the like, big & bulky, reputable/eminent.
shahr n.m. (pl. shuhur and ashur) 2:185, 2:185, 2:194, 2:194, 2:197, 2:217, 2:226, 2:234, 4:92, 5:2, 5:97, 9:2, 9:5, 9:36, 9:36, 34:12, 34:12, 46:15, 58:4, 65:4, 97:3
LL, V4, p: 336, 337 ##http://ejtaal.net/aa/#q=shhr