A neat idea, however an eccentric orbit can't be tidally locked.

If you consider what it means to be tidally locked:

Tidal locking (or captured rotation) is when one side of an astronomical body always faces another. It is also called synchronous rotation.

So we want a planet which always faces the star, to do this we need our planet to rotate at the same rate it moves around the star. Consider the image below:

Between points a and b our planet will move more quickly than between points c and d despite covering the same angular distance. (This can be seen in an animation from the wiki page). So the planet would need to vary it's rate of rotation to speed up from a to b and rotate slower between c and d. Unfortunately there are no reasonable mechanisms to effect the rate of rotation in this way.

A neat idea, however an eccentric orbit can't be tidally locked.

If you consider what it means to be tidally locked:

Tidal locking (or captured rotation) is when one side of an astronomical body always faces another. It is also called synchronous rotation.

So we want a planet which always faces the star, to do this we need our planet to rotate at the same rate it moves around the star. Consider the image below:

Between points a and b our planet will move more quickly than between points c and d despite covering the same angular distance. (This can be seen in an animation from the wiki page). So the planet would need to vary its rate of rotation to speed up from a to b and rotate slower between c and d. Unfortunately there are no reasonable mechanisms to effect the rate of rotation in this way.