Consider that the existence of elementary inverses and the solvability of equations by elementary numbers are two different mathematical problems, as [Chow 1999] points out.
We have $F(z)=A(z,e^z)$. Let's denote the inverse of $F$ by $y$. For the inverse $y$, $F(y(x))=x$ applies. So we get
$$A(y(x),e^{y(x)})=x\tag{1}$$
with $x$ a continuous variable.
Because $y$ is the inverse of $F$, both $F$ and $y$ are injective, and $y$ cannot be a constant function therefore.
1.)
There is a proof in [Liouville 1938] p. 536 - 539 for the equation
$$\log y=F(x,y),$$
where $F$ is an algebraic equation involving both $x$ and $y$. Liouville proves that $y$ is not an elementary function of positive order if $F'_x(x,y),F'_y(x,y)\neq 0$.
(Someone should translate Liouville's works into English.)
2.)
I found that the proof from [Ritt 1948] p. 59 - 62 for Kepler's equation $z-c\ \sin(z)=x\ (c\in\mathbb{C})$ can also be used for the equation $A(z,e^z)=x$. Ritt presents the definition and method of elementary functions of order $n$ of Liouville ([Liouville 1837], [Liouville 1838]).
Ritt's proof goes for the equation
$$z=\log w(x,z)\tag{11}$$
"where $w$, algebraic in $x$ and $z$, involves $z$ effectively. Equation (11) is certainly not satisfied by a nonconstant algebraic $z(x)$."
Ritt then proves that (11) has no solution $z$ that is an elementary function of order $n>0$.
The main theorem in [Ritt 1925], that is also proved in [Risch 1979], implies that our function term $F(z)=A(z,e^z)$ is not in a form that allows to decide if the function $F$ has an elementary inverse or not.
But with the above proofs from Liouville and Ritt, we are able to prove the non-existence of an elementary inverse of $F$.
3.)
For functions $A$ for which equation (1) leads to an equation $P(z,e^z)=0\ $ (2), where $P$ is an irreducible polynomial function of two complex variables with algebraic coefficients, the main theorem in [Lin 1983] states that equation (2) doesn't have nonzero solutions $z$ that are elementary numbers.
It follows with the theorem in Proof Check: Non-existence of the inverse function in a given class of functions, that the function $F$ with $F(z)=P(z,e^z)$ doesn't have an inverse that's an elementary functions.
4.)
Possibly the method of [Rosenlicht 1969] /[Rosenlicht 1976] could also be used to prove the problem from the question.
$\ $
[Bronstein/Corless/Davenport/Jeffrey 2008] Bronstein, M.; Corless, R. M.; Davenport, J. H., Jeffrey, D. J.: Algebraic properties of the Lambert W Function from a result of Rosenlicht and of Liouville. Integral Transforms and Special Functions 19 (2008) (10) 709-712
[Chow 1999] Chow, T.: What is a closed-form number. Am. Math. Monthly 106 (1999) (5) 440-448
[Lin 1983] Ferng-Ching Lin: Schanuel's Conjecture Implies Ritt's Conjectures. Chin. J. Math. 11 (1983) (1) 41-50
[Liouville 1837] Liouville, J.: Mémoire sur la classification des transcendantes et sur l'impossibilité d'exprimer les racines de certaines équations en fonction finie explicite des coefficients. J. de mathématiques pures et appliquées 2 (1837) 56-105
[Liouville 1838] Liouville, J.: Suite du mémoire sur la classification des transcendantes, et sur l'impossibilité d'exprimer les racines de certaines équations en fonction finie explicite des coefficients. J. de Mathématiques Pures et Appliquées 3 (1838) 523-547
[Risch 1979] Risch, R. H.: Algebraic Properties of the Elementary Functions of Analysis. Amer. J. Math. 101 (1979) (4) 743-759
[Ritt 1925] Ritt, J. F.: Elementary functions and their inverses. Trans. Amer. Math. Soc. 27 (1925) (1) 68-90
[Ritt 1948] Ritt, J. F.: Integration in finite terms. Liouville's theory of elementary methods. Columbia University Press, New York, 1948
[Rosenlicht 1969] Rosenlicht, M.: On the explicit solvability of certain transcendental equations. Publications mathématiques de l'IHÉS 36 (1969) 15-22
[Rosenlicht 1976] Rosenlicht, M.: On Liouville's theory of elementary functions. Pacific J. Math. 65 (1976) (2) 485-492
