You are quoting expected CR - the final CR is much higher.
On base hp 178 × 1.5 for immunities gives 262 - so defensive CR 13. Expected AC is 18, so an AC of 16 gives no adjustment.
Expected damage per round (~45-55) from its breath weapons (3 with a 1/3 recharge chance - it will have one every round) plus some acid blood splash against 1 PC - assume it can hit 2, so 90-110 or Offensive CR 16. This is an "effects" monster so any adjustment to this should be on its save DC which is 16 instead of the recommended 18 so no change there.
Averaging and rounding up gives a CR 15.
At least one of them, if you consider older edition info
This is partially drawn from some older information, but here you go...
As you stated, the Monster Manual says that Kuo-Toa 'create' gods by believing in them so hard that they manifest. It continues to say this...
One of the most revered gods of the kuo-toa is Blibdoolpoolp the Sea Mother, who takes the form of a female human with a crayfish head, a crayfish’s claws, and an articulated shell covering her shoulders. Blibdoolpoolp was likely invented by a kuo-toa that improved on a broken human statue by adding the limbs and head of a crustacean. In sudden awe of its handiwork, it then named the resulting form a god.
While no 5E information exists expanding on this, there is some 3E material that covers this particular goddess.
From the 3rd Edition book Faiths and Pantheons, we have a table on page 221 (list of non-human deities) that has record of Blibdoolpoolp. She is listed as an Intermediate Deity, thus being on the same power level as gods such as Kelemvor, Tymora, Lloth (Lolth), and Helm. She has the Domains of Destruction, Evil, and Water (Pared back to just 'Death' in 5E, see Appendix B of the DMG.)
The 3rd Edition book Player's Guide to Faerun gives us a little more details. On page 53, it introduces her Home Plane: The Fated Depths, which she shared with Sekolah, the god of the Sahuagin.
The Drowning Goddess, on the other hand, does have a realm—a spherical temple that resembles a small moon drifting through the unholy waters of the plane.
So, as we can see from here...Blibdoolpoolp is spoken of in the same breath and in the same manner as all other gods. Apart from the fact that she was created by her worshipers believing in her hard enough, she appears to be a fully functional deity.
Now, as for any other gods they may invent...the books are less clear. The phrasing makes it clear that a newly invented god has to reach a critical mass of worshipers before it would be possible for them to actually exist, but specific details are never provided.
Of note, The Fated Depths apparently ceased to exist during the Spellplague. We have no updated information from 4E or 5E on where Blibdoolpoolp presently resides.
Best Answer
It wasn't in the errata, so yes it is accurate.
Lacking any corrections from the game designers themselves, CR1 is the correct CR for the Kuo-toa Whip.
You can reconstruct the creature's CR using the rules in Creating a Monster in the DMG page 273.
Using those rules, we get that the Whip has a Defensive CR of 1/4 (1/2 if we assume it always has Shield of Faith on), and an Offensive CR of 1. Its spellcasting doesn't really matter for Offensive CR as it has little damaging spells. So it should really only be CR 1/2.
But a critical thing you're missing is that the ultimate Challenge Rating for a monster isn't derived from crunching numbers, but rather through playtesting again and again.