The way I understand your description, the motion of the light and the spring tension are perpendicular to the direction of acceleration. Consider the problem in the (instantaneous) rest frame of the mirrors and spring.
Let's use coordinates where x is the direction of the photon's motion (in the unmoving rest frame), and y is in the direction of acceleration (and motion) of the mirrors.
The relevant equations are spring force $F = kx$, photon momentum in the mirror (x) direction = $P_x = \hbar k_x$, photon rate $1/T$ where $T$ is the time between photons, and the photon force = change in momentum per unit time = $2\hbar k_x / T$. The requirement for no change to the mirror is that $kx = 2\hbar k_x/T$.
Clearly the spring is just a spring in the moving rest frame and so there's no change to the spring force $kx$.
As for the photons, their momentum perpendicular to the direction of velocity (or acceleration) is unchanged, so there's no change to the momentum any single photon imparts to the mirror $2\hbar k_x$. Uh, note that since the photon reverses its direction, it imparts twice its momentum to the mirror. Another way of saying the same thing is to note that stationary and moving observers agree on the number of photon wavelengths $\lambda_x$ between the two mirrors. They also agree on the distance between the two mirrors. Therefore they agree on the wave number $k_x = 2\pi/\lambda_x$ for the photon's momentum in the x direction.
So what's left is the rate at which the photons impact the mirror $1/T$. This rate does not depend on the acceleration (relativity problems rarely do). Instead it depends on the velocity of the mirror.
As the mirror velocity increases, the distance traveled by the photons in the moving frame increases. Thus the time between photons increases and the photon force decreases.
Of course in the unmoving rest frame the photon rate is unchanged, it is only in the moving frame that the photon rate changes. This effect is called time dilation. Consequently, the non moving observer, on noting the force on the mirror, must conclude that the spring constant $k$ has changed due to the motion of the spring. For a discussion of this interesting effect, see: http://www.mathpages.com/home/kmath068/kmath068.htm