This problem seems more suitable for an OGF than an EGF.
Using $z$ for any letter $A,B$ or $C$ and $w$ for $B$ or $C$ we obtain
$$(1+zw+z^2w^2+\cdots)
\left(\sum_{q\ge 0} z^q (zw+z^2w^2+\cdots)^q\right)
(1+z).$$
This is
$$\frac{1}{1-zw} \frac{1}{1-z\times zw/(1-zw)} (1+z)
\\ = \frac{1+z}{1-zw-z^2w}.$$
Now for $w$ we have two choices and we finally obtain the OGF
$$\frac{1+z}{1-2z-2z^2}.$$
As a quick check the DFA
method produces
> GFNC([[0,0]], 3, true);
[[0, 0]]
Q[], 0, Q[0]
Q[], 1, Q[]
Q[], 2, Q[]
Q[0], 0, Q[0, 0]
Q[0], 1, Q[]
Q[0], 2, Q[]
Q[0, 0], 0, Q[0, 0]
Q[0, 0], 1, Q[0, 0]
Q[0, 0], 2, Q[0, 0]
z + 1
- --------------
2
2 z + 2 z - 1