Csdrhrt

In the book Ziegler, Tent: A course in model theory it states

Exercise 5.5.7. Let $T_1$ and $T_2$ be two model complete theories in disjoint
languages $L_1$ and $L_2$. Assume that both theories eliminate $exists^infty$ . Then $T_1cup T_2$ has a model companion.

I want to prove this statement. Sine $T_1cup T_2$ also eliminates $exists^infty$ (i.e. in all models realisation sets have either a finite upper bound or are infinite) my strategy was to use the previous exercise

Exercise 5.5.6. Assume that T eliminates the quantifier $exists^infty$. Then for every formula $varphi(x_1 , . . . , x_n , overline{y})$ there is a formula $theta(overline{y})$ such that in all models $mathfrak{M}$ of
$T$ a tuple $overline{b}$ satisfies $theta(overline{y})$ if and only if $mathfrak{M}$ has an elementary extension $mathfrak{M}'$ with
elements $a_1 , . . . , a_nin M'setminus M$ such that $mathfrak{M}vDashvarphi(a_1,...,a_n)$.

I define the theory

$(T_1cup T_2)^*:={theta;|;$there exists a $L_1cup L_2$-formula $varphi(x_1,..x_n)$, that is satisfiable in all models $mathfrak{M}vDash T_1cup T_2$ such that $mathfrak{M}vDashthetaLeftrightarrow$ there exists an elementary extension $mathfrak{M}'$ with
elements $a_1 , . . . , a_nin M'setminus M$ such that $mathfrak{M}vDashvarphi(a_1,...,a_n)$$}$

To me it seemed to be right choice for the model companion of $T_1cup T_2$ until I tried to prove that $(T_1cup T_2)^*$ is model complete (without success). Does this choice make sense?