Does the entropy of the universe change as expansion exceeds the speed of light?

If the whole universe is homogeneous, it will be as likely to loose a photon than to gain one that come from the other side, so on average the entry does not change. However, we are actually in an accelerating universe with accelerated rate of expansion. In this case the observable universe actually diminishes in size, and so the entropy. The total entropy of the observable universe only grows if it is a closed system. The visible universe of a particular observer, does not define any physical border, just a causal/observable one. The entropy inside the observable universe (and its radius) does not grow, and actually, by the holographic principle, the entropy gets actually reduced. This is because of the reduction of mass and energy and of available microstates as the radius diminishes. The extreme case will be in the future if the big rip happens, every fundamental particle and every black hole will become isolated from the rest of the universe, so each disconnected part of the universe will in fact have very low entropy (although the entropy of the whole universe will grow)

The Second Law of Thermodynamics states that the state of entropy of the entire universe, as an isolated system, will always increase over time. The second law also states that the changes in the entropy in the universe can never be negative. Put another way as the universe increases in volume there are more ways to arrange the mass of the universe in more configurations. Therefore as the size and volume increase at an accelerated rate the potential for even more configuration of mass increases. This is true regardless of the speed of light.

The speed of light is relevant when we examine the observable cosmic universe or hubble horizon -

One can define a so-called "Hubble Horizon" which shows roughly how far light would travel if space were not expanding. This size is

$chi = c t$

where t is the lookback time since the Big Bang (otherwise known as the age of the universe) which, according to the Friedmann Equations, is:

$t = int^{a}_{0}{frac{da}{H_0 sqrt{Omega_R a^{-2} + Omega_m a^{-1} + Omega_k +Omega_Lambda a^2}}}$

where $H_0$ is the Hubble Constant and the $Omega$ density parameters are, in order, the density of radiation, matter, curvature, and dark energy scaled to the critical density of the universe.

Today, roughly:

$chi_0 = frac{c}{H_0}$

yielding a Hubble horizon of some 4.2 Gpc. This horizon is not really a physical size, but it is often used as useful length scale as most physical sizes in cosmology can be written in terms of those factors.

You ask -

The potential encoded information in a photon that is at the edge of the observable universe would seem to be lost as the universe expands. Does that loss of information contribute to the overall entropy of the universe or is the information fully encoded in our measurement apparatus just before the photon is lost?

Very much doubt that it would be encoded in the measuring apparatus. And yes since it is no longer observed in the local Hubble Horizon or sphere the data would appear to be lost locally and a net local increase in entropy could said to take place. All the while as the entropy of the universe is increasing at an increasing rate along with the exponentially increasing volume of the universe. Basically just a subset of the whole.

For a vast over simplification - take the universe to be analogous to a very well insulated piston containing a gas. If you expand that volume normally it will be adiabatic / constant entropy - no heat loss by definition of it being well insulated.

Now add in the accelerating expansion of the universe such that regions relative to us are no longer accessible because they are moving away from us faster than the speed of light. In the piston analogy it would be as if we've inserted a thermal barrier between sections of the piston that do not allow transfer of heat. Inserting these barriers does nothing to the entropy of the piston. Therefore for the universe as a whole there is no change in entropy.

Now if you're interested in the entropy of the accessible universe, that has certainly decreased, but so has the amount of matter/energy.