How many photons are there going to be? The ship (or warp bubble or whatever) is only collecting photons from distant stars; Effectively it's surrounded by a black body at ~3K. How much energy is it going to collect in 4ly of travel?
I would guess that the vast majority of these photons would come from the destination star, 'cos it's the only one you get close to; with the rest of the universe adding a trivial fraction. So you are looking at what, same amount of energy as the solar (? centaurar) energy incident on the ship if it was at the destination for some fraction of the 4 year output? It's going to be a small fraction too, with the inverse square law - most of the trip is spent a long way away from any stars, including the target system.
Maybe not.
As I see it, in the case of the Alcubierre Drive, if you're not careful the drive would sweep nearly all photons in front of the ship between the Earth and Alpha. These photons would include the photons from the Sun and from the three stars in Alpha Centauri, and then some, and that's photons emitted by those stars
during 4.4 years. If you're subjected to that amount of photons, I'm sure you're burnt to a crisp, like being exposed to the equivalent of 4.4 years of sunlight, unscreened by any atmosphere, within potentially one small fraction of one second. Crisp.
EB