Actually if the neutron star is in the center then the added gravity in the layer near it will cause greater fusion, when increases pressure causing the layer to expand outward. As it expands it encounters the next layer, which will absorb the energy and begin the same process on a smaller scale. So forth untill it reaches the photosphere. In the mean time the inner layer has transfer excess momentum, has cooled some from it's expansion, it rate of fusion decreases and it collapses to the point it was before, actually a little less. Then it begins the process again. The result is a slightly variable vibrating star. I wonder is it might be possible to create sympathetic vibrations that way. Probably not, It would be most interesting.
There is however one major flaw in the above model. Supposedly the neutron star was once part of binary system with the red giant. It is not suddenly going to appear in the core. Instead, it is going to slowly absorb material until it's orbit "decays" into one below the photosphere. I don't know if parts of star will be eject when gases impinge. Certainly tides on the star will help. ( Also the neutron star will generate tides on the giant affecting it's weather. )
So the neutron star will act like a giant stirring, slowly stirring up layers of the gas. The romantic in me would like to think this is going to generate some massive and quite interesting effects. The realist though says that any effects will probably be small and hardly noticeable after a few millennium.