Since the tidal bulge is always in the same place, how would that affect ocean tides? Would they change throughout an elliptical orbit, due to changing distance from the star? How exactly would they behave? And would tides on a tidally locked moon behave the same way?
Since the tidal bulge is always in the same place, how would that affect ocean tides?
The concept of tidal bulge is a useful fiction, but fiction nonetheless. For an object in an eccentric orbit, the object's rotation rate and the object's orbital rate are rarely equal. There would be tidal effects. We see this on the Earth's Moon, where eccentricity results in moonquakes.
We also see this on Jupiter's moon Io. Io is caught in a 1:2:4 resonance with Ganymede and Europa. These resonance effects tend to make Io's orbit more eccentric. This increased eccentricity in turn increases the tidal effects by Jupiter on Io. (Io's tides would be frozen if Io was in a circular orbit.) The increase in tidal effects in turn causes internal heating in Io. This in turn results in factors that decrease Io's eccentricity. The end result is a nice hysteresis loop. Io's core is coolish and hard when it is in a fairly circular orbit. Ganymede and Europa pull it out this circular orbit. Tidal heating due to increased eccentricity makes Io's core get warmer and not quite so hard, which pulls it back to more circular orbit. And so on.
With an elliptical orbit that is "locked" to the parent the orbital velocity will change--as seen from the child the parent will move backwards during the close part of the orbit and forward during the far part of the orbit. Thus you will have tides, but they'll be small and go back and forth, not around.
Note that even with a circular orbit you can still have wobble causing the same sort of thing.
Eventually the tides will circularize the elliptical orbit and damp the wobble of the circular orbit but that takes time. It very well might not have happened by the time the star dies.