I'm looking for an analytical solution to the equation $$\frac{\sin n\theta}{\sin m\theta}=a \quad (n,m\in\mathbb Z)$$ where the constant $a$ is real and can be both positive and negative.
The solution is needed for the unknown $\theta$; the rest of the parameters are known.
One approach I took was to write the sine functions as $\sim (e^{i x} - e^{-ix})$ which resulted in a polynomial equation of degrees set by $n$ and $m$, with no apparent analytical solution.
Is there an alternative route that can be taken, even by using special functions?
You said you have attempted to convert your issue into a polynomial equation. But do you know that there already exists "on the shelf" Chebyshev polynomials of the second kind $U_n$ :
$$\displaystyle U_{n-1}(\cos \theta )=\frac {\sin n \,\theta}{\sin \theta }$$
giving
$$U_{n-1}(x)=a U_{m-1}(x)$$
which is a $\max(n,m)-1$ polynomial equation in $x:=\cos \theta$, keeping the roots whose absolute values are at most $1$.
Therefore, your question has essentially a negative answer: looking for an explicit analytical formula is elusive, because in general, above degree 4, most polynomial equations do not have "formulas" for their roots...
Remark 1: connection with hypergeometric function $ {}_{2}F_{1}$ (you find it in the Wikipedia article):
$$U_{n}(x)=(n+1)\ {}_{2}F_{1}\left(-n,n+2;{\tfrac {3}{2}};{\tfrac {1}{2}}(1-x)\right)$$
Remark 2: A completely different approach would be to write your equation under a form involving the cardinal sine function $\operatorname{sinc}$ i.e.,
$$ \frac{\operatorname{sinc}(n \theta)}{\operatorname{sinc}(m \theta)}=\frac{n}{m} a$$