Finding two languages satisfying conditions
A slight improvement:

$L_2 = E_{TM} \backslash L_1 $ (since we require that $L_1 \cap L_2 = \emptyset$).

$L_1$ must be a language of turing machines, i.e. $L_1 = \{\langle M\rangle \mid \dots\} $, so that $L_1 \cup L_2 = E_{TM}$.

So let's set $L_1 =$ $\{\langle M\rangle\mid  C_0\text{ is a reject state}\}$.

The following holds:


$L_1 $ is infinite and decidable (easy to decide given $M$, what the configuration of the first state is).
$L_2 $ is infinite and unrecognizable.