How is this difference reconciled? Is there something I missed about Nimble Escape?
It isn't. No.
It has been said before on this site (but damned if I can find it) that the CR of the monsters in the Monster Manual were not set using the methodology set forth in the Dungeon Master's Guide - there were established by eyeballing and playtest.
Notwithstanding, the DMG does say (p. 237):
Creating a monster isn't just a number-crunching exercise. The guidelines in this chapter can help you create monsters, but the only way to know whether a monster is fun is to playtest it. After seeing your monster in action, you might want to adjust the challenge rating up or down based on your experiences.
See Ogre CR calculation: is it wrong or am I missing something? for a calculation that goes the other way.
Think of the CR calculation as trying to work out how to fly a spacecraft to the moon with calculations on the back of a napkin.
When the Monster Manual's Manticore (page 213) uses its Multiattack, it can either combine a bite and two claw attacks, or it can make three tail spike attacks. A bite and two claws comes out to 19 damage, on average, and three tail spikes deal a combined 21 damage, on average. The DMG tells us to make CR judgments based on the more damaging option available to a monster, so the Manticore gets to use 21, but our spikeless Dragonne is limited to 19.
Looking at the Monster Statistics by Challenge Rating table (DMG 274), 21 damage per round is in the range of a CR 3 monster. 19 damage per round is in the range of a CR 2 monster. The cutoff is 20/21, so this is a fairly fine distinction we're making.
However, as described under "Final Challenge Rating" (on the same page), monster CR depends on other factors. The Manticore's HP falls in the CR 1/2 range, and its AC is in the CR 4 range, so its Defensive Challenge Rating comes in at 2. Our Dragonne's new damage per round is in CR 2 range, and its attack bonus is in the CR 4 range, so its Offensive challenge Rating is 3.
We get the final average rating by calculating the average of the monster's offensive and defensive challenge ratings: 2.5. "Round the average up or down to the nearest challenge rating," the DMG says. According to the text, we're free to treat the Dragonne as a CR 2 or a CR 3 monster.
But all this math only confirms what was pretty obvious in the first paragraph: Numbers-wise, we haven't changed the Manticore very much. We've decreased its average damage per round by 2, which in many cases wouldn't be enough to affect the CR at all.
Removing the Manticore's tail spikes may be an important change strategically: The Manticore is (roughly) as dangerous at range as it is in melee, but our Dragonne doesn't have a ranged attack. The DMG's guidelines for monster creation and CR calculation don't say anything about the difference between ranged and melee attacks, though, which means they don't affect Challenge Rating, the abstract, mathematically-defined concept, at all. Whether the Dragonne's reliance on melee has a calculable effect on how challenging it is in "real life" will come down to how you use it.
Best Answer
Permanent invisibility increases Challenge Rating by 3.
Assuming (dis)advantage is ±5 to a roll, CR increases by 2 before accounting for the invisibility being permanent.
The conventional wisdom is that advantage and disadvantage roughly translate to a ±5 to a d20 roll (for more details see the last section of this answer). Taking this rough assumption, we can apply it to the Challenge Rating guidance found in the Dungeon Master's Guide:
Assuming then that having disadvantage in attacks against the creature correlates to +5 Armor Class, and the creature having advantage on all attacks correlates to +5 to the bonus to hit, the result is a +2 to both Defensive and Offensive Challenge Rating, and since overall Challenge Rating is the average of these, we net a +2 to overall Challenge Rating. However, this is before we account for the permanency of the invisibility.
+2 CR is consistent with guidance given for similar features, but I suggest +3 because permanent invisibility is superior to these features.
Unfortunately, the Challenge Rating system doesn't encode every property of a creature. Fortunately, we have some guidance given for similar features in the Monster Features Table. The Goblin has the Nimble Escape feature, and the guidance given in the table matches the intuition I gave above:
This feature allows a goblin to Hide as a bonus action each around, which can result in similar circumstances as permanent invisibility - advantage on attacks by and disadvantage on attacks against the creature:
This nets a +2 to Challenge Rating for the reasons described in the first section.
However, taking the Hide action requires rolling a Dexterity (Stealth) check, and so is not guaranteed to work every time. Permanent invisibility is guaranteed to work, so it is clearly superior to Nimble Escape. Additionally, permanent invisibility also provides the other potential benefit of Nimble Escape as it prevents opportunity attacks; the rules for opportunity attacks state:
Therefore, I suggest +3 to CR, instead of +2.