I don't know if such an eccentricity is possible given a tidally locked planet. Why ? Because it would mean an irregular planetary rotation. A tidally locked planet is still rotating, the thing is, its rotation has the same length as a year.
If you have too much eccentricity, the planet can't always face its sun from the same side. If this was the case, while away from its sun the planet will spend much more time facing one direction, and then fall down to the perigee at great speed, thus rotating faster to keep the sun in view. That's not going to happen.
The big question is : where is your zone of habitability ? The easiest and most probable is on the "twilight ring". If so, see answers to Planet Tidally-Locked to its star having eclipse day/night cycles?
If your planet is habitable everywhere (still quite possible with a good enought weather system), then by definition a tidally-locked planet will have a hard time getting day/night cycles... Unless...
Remember the problem with eccentricity ?
If you have too much eccentricity, the planet can't always face its sun from the same side. (quoting from the same answer, check)
Then you'd have a "day equals year" planet, not quite tidally locked in the usual sense, but with a wiggle proportionnal to its eccentricity. See the Moon libration as an exemple, but with greater wiggling.
Libration in longitude results from the eccentricity of the Moon's orbit around Earth; the Moon's rotation sometimes leads and sometimes lags its orbital position.
Some quarter (more or less, given the eccentricity) of the planet will always be day (with the sun still varying in height), the opposite quarter always night, and the others will have that day/night cycle.
Sadly, I have no simulation to offer you right now, but I guess it could be an interesting day/night pattern, adding the fact that for some longitudes the day will be that of a dim sum (at the apogee), while for others the day will se the sun at its closest (at the perigee).