Yes, we can. It seems to me a locus finding problem in reflection geometry. Incident ray direction and incident ray segment length can be taken $2 \theta$ anti-clockwise and constant $b$ respectively when reflected ray QR is horizontal.
Directly from the figure at left parametric equation of locus of P of two components from $(a=1,b)$ is the spiral
$$(x_P,y_P)=(a\cos\theta_Q+b\cos 2\theta,\,a\sin \theta_Q + b\sin 2\theta_Q)\tag 1$$
which ( appearance similar to Limacon ) is an algebraic curve of fourth degree (not converted here to cartesian coordinates ) sketched below:
When $b$ segment length is perturbed the yellow band is obtained with rulings of all possible incident rays.
For Ruler and Compass choose a point R, draw a line parallel to x-axis to cut mirror M at Q which is the unique point. Drawing a perpendicular to OQN, transferring angle $\theta$ to left of normal OQN by cutting off arc length QR with compass to find P, is simple construction.
If the above is an indirect method we can directly find coordinates of Q:
Using above parametrization and from the condition that the angular bisector of QP, QO is QR we arrive at the required unique $\theta_Q $ implicit condition to be satisfied for given P$(x_P,y_P).$
Calculation detail is omitted.
$$y_P \cos 2 \theta_Q- x_P \sin 2 \theta_Q + a \sin \theta_Q =0 \tag2 $$
Numerical solution by Newton-Raphson can determine $\theta_Q$.