How to solve for $\theta$ in terms of $a,b,c$ in this expression?

Considering the first equation $$c = \frac{2a}{\sin (\theta)}(1-\cos (\theta))+\frac{b\,\sin (\theta)}{\cos (\theta)}$$ using the tangent half-angle substitution, we end with $$c=\frac{2 t \left(-a t^2+a+b\right)}{1-t^2}$$ that is to say $$2 a t^3-c t^2-2 t (a+b)+c=0$$ that you can solve using Cardano method.

Probably easier would be Newton method for solving the cubic equation. Starting using $t_0=0$, we should have $t_1=\frac{c}{2 (a+b)}$ and $$t_2=\frac{c^3 (b-a)+4 c (a+b)^3}{8 (a+b)^4-2 c^2 (a-2 b) (a+b)}$$ $$t_{n+1}=\frac{4 a t_n^3-c t_n^2-c}{2 \left(3 a t_n^2-c t_n-(a+b)\right)}$$

Similarly, if $\theta$ is small, we could expand the first equation as Taylor series $$c= (a+b)\theta+\frac{a+4 b}{12} \theta ^3 +\frac{a+16 b}{120} \theta ^5 +\frac{17 (a+64 b)}{20160}\theta ^7+O\left(\theta ^9\right)$$ and use series reversion to get $$\theta=\frac{c}{a+b}-\frac{ (a+4 b)}{12 (a+b)^4}c^3+\frac{ \left(a^2+2 a b+16 b^2\right)}{80 (a+b)^7}c^5+O\left(c^7\right)$$

Edit

Still assuming small values fo $\theta$, we could build at $\theta=0$ the $[2,2]$ Padé approximant and get for the whole $$f(\theta) = \frac{2a}{\sin (\theta)}(1-\cos (\theta))+\frac{b\,\sin (\theta)}{\cos (\theta)}-c$$ the approximation $$f(\theta)\simeq\frac{-c+ (a+b)\theta+\frac{c (a+4 b)}{12 (a+b)}\theta ^2}{1-\frac{(a+4 b)}{12 (a+b)} \theta ^2 }$$ Solving the quadratic $$\theta\simeq \frac{2 \left(\sqrt{3 c^2 (a+4 b) (a+b)+9 (a+b)^4}-3 (a+b)^2\right)}{c (a+4 b)}\tag 1$$ which seems interesting.

Let us try it for $a=1$ and $b=4$. Give $\theta$ a value to compute $c$ and recompute the estimate $\theta_*$ from $(1)$. The table below reproduces results (in degrees). $$\left( \begin{array}{ccc} \theta & c & \theta_* \\ 5 & 0.43728 & 5.00001 \\ 10 & 0.88029 & 10.0003 \\ 15 & 1.33510 & 15.0020 \\ 20 & 1.80853 & 20.0082 \\ 25 & 2.30862 & 25.0249 \\ 30 & 2.84530 & 30.0617 \\ 35 & 3.43143 & 35.1324 \\ 40 & 4.08434 & 40.2567 \\ 45 & 4.82843 & 45.4605 \\ 50 & 5.69963 & 50.7782 \\ 55 & 6.75373 & 56.2542 \\ 60 & 8.08290 & 61.9466 \end{array} \right)$$

Update

After comments, it is quite sure that we can make better knowing that the area of interest is around $\theta=\frac \pi 4$. To stay "simple", writing the $[1,1]$ Padé approximant, we should get $$\theta\simeq \frac \pi 4+ \frac{\alpha +\beta c}{\gamma+\delta c}$$ where $$\alpha=2 \left(\left(6 \sqrt{2}-8\right) a^2+\sqrt{2} a b+b^2\right)$$ $$\beta=2 \left(\sqrt{2}-2\right) a-2 b$$ $$\gamma=2 \left(\sqrt{2}-2\right) a^2+3 \left(5 \sqrt{2}-8\right) a b-2 b^2$$ $$\delta=\left(4-3 \sqrt{2}\right) a-2 b$$ Applied to the case $$a= \pi 160\frac{12}{360}=\frac{16 \pi }{3}\qquad b= \pi 160\frac{50}{360}=\frac{200 \pi }{9}\qquad c=80$$ this would give $$\theta\simeq \frac \pi 4 +\frac{90 \left(6 \sqrt{2}-37\right)+\left(337+366 \sqrt{2}\right) \pi }{\left(1161 \sqrt{2}-2497\right) \pi -90 \left(13+9 \sqrt{2}\right)}\approx 0.761710$$ which converted to degrees would give $43.6427$ while the exact solution would be $\theta=0.7617146$ corresponding to $43.6430$. Not too bad.

Keeping $a=\frac{16 \pi }{3}$ and $b= \frac{200 \pi }{9}$ and varying $c$ over a quite large range, here are some results. $$\left( \begin{array}{ccc} c & \theta_{approx} & \theta_{exact} \\ 60 & 35.1540 & 35.2569 \\ 65 & 37.4779 & 37.5244 \\ 70 & 39.6591 & 39.6759 \\ 75 & 41.7103 & 41.7142 \\ 80 & 43.6427 & 43.6430 \\ 85 & 45.4666 & 45.4665 \\ 90 & 47.1906 & 47.1894 \\ 95 & 48.8228 & 48.8165 \\ 100 & 50.3704 & 50.3528 \end{array} \right)$$

New update

We could still do much better at the price of a quadratic equation. Around a given angle $\theta_0$, develop the rhs of the original equation as a $[2,2]$ Padé approximant and solve for $\theta$. I shall not give here the formulae but just the results for the last worked case $(\theta_0=\frac \pi 4, a=\frac{16 \pi }{3}, b= \frac{200 \pi }{9})$ $$\left( \begin{array}{ccc} c & \theta_{approx} & \theta_{exact} \\ 60 & 35.256535 & 35.256856 \\ 65 & 37.524331 & 37.524418 \\ 70 & 39.675901 & 39.675917 \\ 75 & 41.714232 & 41.714233 \\ 80 & 43.643029 & 43.643029 \\ 85 & 45.466540 & 45.466540 \\ 90 & 47.189400 & 47.189400 \\ 95 & 48.816481 & 48.816478 \\ 100 & 50.352781 & 50.352763 \end{array} \right)$$