One possibility is an object that has an extremely eccentric orbit with a perihelion (closest approach to the star) not too far outside the tidally locked planet's orbit. And a resonant orbital period. Thus the object might complete exactly ten or a hundred or some other number orbits for every one the tidally locked planet does.
The outer object with the eccentric orbit would travel faster the closer it was to the star as it got closer to the star. So when it was at perihelion it would be moving very fast as it swung around the star with enough excess speed to travel far back out again. And the duration of it's closest flyby of the star might thus be about the same as the period of the tidally locked planet's year/day.
The light of the planet's star scattered around the daytime sky will drown out the light of most celestial bodies except the star. Especially if the atmosphere is more dusty or foggy or opaque on the sunward side of perpetual daytime.
On Earth, objects visible in the daytime sky include:
1) the Sun, apparent magnitude -26.74 (the higher the magnitude the lower the brightness)
2) the Moon, apparent magnitude -12.90 at full moon, down to -2.5 at new moon.
3) comet Ikeya-Seki (1965) maximum apparent magnitude -10.00. The brightest comet to be visible during the daytime.
4) Iridium satellites. Maximum apparent magnitude of flares -9.5.
5) Supernova seen in 1006. Maximum apparent magnitude -7.5.
6) Crab supernova seen in 1054. Maximum apparent magnitude -6.00.
7) International Space Station. Maximum apparent magnitude -5.90.
8) planet Venus Maximum apparent magnitude -4.90, minimum -3.2.
Objects with apparent magnitude less than -4.0 are not visible when sun high in sky.
9) planet Jupiter. Maximum apparent magnitude -2.94, minimum -1.61.
10) planet Mars. Maximum apparent magnitude -2.91, minimum +1.84.
Objects with apparent magnitude less than -2.5 are not visible in daytime sky even when the sun is less than 10 degrees above the horizon.
11) planet Mercury. Maximum apparent magnitude -2.45, minimum +5.7.
12) star Sirius. Apparent magnitude -1.47. All other stars and planets are not as bright as Sirius.
There are about 9,096 stars visible in dark skies to average human eyes from Sirius at -1.47 down to stars at apparent magnitude +6.5.
Note that it is possible to see a few of the brighter stars and planets at twilight. Perhaps the nightsiders rule the twilight zone and keep the daysiders out of it.
The tidally locked planet would not have a moon. The author can decide if the daysiders have any artificial satellites bright enough to be seen in the day. Comets bright enough to be seen in the day are rare in our solar system, and supernovas visible in the daytime are much rarer in our region of the galaxy (which is one of the necessary conditions for life on Earth). The author can decide if those conditions are different in his solar system.
The author can decide if there will be any stars bright enough and close enough to be seen in the daytime of his planet. The author can decide if there are any planets in the solar system that sometimes get bright enough to be seen in the day. In short the author can ensure that the only celestial object the daysiders ever see in the sky (except for rare and unpredictable events like supernovae and daylight comets) is their sun.
The nightsiders will see the stars all the time and count time by the time it takes for a particular star to rise above the horizon, reach it's highest point in the sky, and set below the horizon, and repeat. They will know that the outer celestial body will take a specific number of such periods to return to sight.
So the only problem is to make sure that the outer celestial body is not bright enough to be seen in the daylight sky until it passes below the horizon as seen from the dayside.
You may need to have someone run some computer simulations to find a set of orbits that makes the outer celestial body only visible when passing by the night side.
It is possible that as the outer object passes close to the night side of the planet it suddenly flares up briefly in brightness due to developing a coma like a comet or having a frozen atmosphere that becomes vapor and makes the body much more reflective.