Carréphylic ancestors
Let 'x' be carréphobic, 'y' carréphylic and 'a' a natural number such that x = a2y ⇔ √x = a√y.
Since y is carréphylic, the approach of √x should pose no practical problems.
Here's how to find the 's' and 'p' of the rational approach of x:
s2-xp2 = 1 ⇔ s2-y(ap)2 = 1.
The fraction s/ap is the ath fraction of the set of sp-blocks of the rational approach of y, starting from the trivial block.
It is therefore the first fraction of the base-a series accelleration of the rational approach of y.
The corresponding factor '2s' is Fa or the factor of y 'base-a'.
Since √x = a√y the sp-fraction of x is a(s/ap) = s/p.

Example carréphylic ancestry
The position of a fraction indicates whether it is over or under the root-value.

The approach of √7 - carréphylic

The approach of √28=2√7 - carréphobic

Using the above data we find that the first non trivial sp-fraction of 28 is twice the second non-trivial fraction of 7, or 127/24.
The corresponding factor 2*127 is the base-2 accelleration of the factor of 7: 162-2.

The approach of √63=3√7 - carréphylic

63 is one less than a square, so the exception mentioned in on root approach applies: 127 and 16, as rendered by the formula, are not the first non-trivial sp-block, but the second, the first being 8 and 1 because 82-63*12 = 1 satisfies the diophantine equation.
Nevertheless, trice the third non-trivial fraction of 7 is 2024/255 and the resulting series is the base-3 accelleration of the sp-series of √63.
The corresponding factor 2*2024 is the base-3 accelleration of the factor of 7: 163-3*16.

The approach of √112=4√7 - carréphobic
As it happens 112 falls outside the first 100 non-squares, so I didn't do a search, but using the above data the first non trivial sp-fraction of 112 is four times the fourth non-trivial fraction of 7, or 32257/3048.
The corresponding factor 2*32257 is the base-4 accelleration of the factor of 7: 164-4*162+2.

Carréphylic descendants
Let 'x' be carréphobic, 'y' carréphylic and 'a' a natural number such that a2x = y ⇔ a√x = √y.
Here's how to find the 's' and 'p' of the rational approach of x:
For y the first non-trivial 's' and 'p' satisfy: s2-yp2 = 1 ⇔ s2-x(ap)2 = 1 ⇔ 's' and 'ap' are the first non-trivial element of the set of sp-blocks of the rational approach of x. The corresponding factor is the same.

Example carréphylic descendency

The approach of √13 - carréphobic
The fractional approach of root 13, the first carréphobic number, as provided by Ed's straightforward program, looks like this:

As it happens, 52*13 = 325 = 182-1 and thus carréphylic. The formula renders 649/36 as the first sp-block of the approach of 5√13, making 649/180 the first sp-block of √13 with the same factor 1298.
Easy as fruitpie.