To make anything like an "accurate" guess, you need a lot more detail (e.g., size and location of land masses and oceans, actual distance between luminaries, etc.). But, even without that some general principles can be worked out using analogies to earth and to so-called eyeball planets.
Based on the work that's been done modeling climate on eyeball planets, it does look like you will get your pseudo-Hadley cells. (Note that tidally locked planets do have some Coriolis effect but it's usually modeled as significantly weaker than earth's so I'm using it as a stand-in.)
The question specifies that the climate directly under the luminary is similar to that at earth's equator. This suggests that we can use earth as a model for our Hadley cells.
On earth, the Hadley cells run about 30 degrees north/south of the equator. However, that is probably impacted by earth's axial tilt (the outer edges of the tropics are a lot more "tropical" during the local summer). Let's be conservative and shrink our cell by about 1/3, so 20 degrees which works out to about 2200km.
Our Hadley cell runs from directly under the luminary out for about 2200km. But how big is our tropical climate zone? (Note that this isn't the same as the "tropics" since we're interested in climate, not orbital mechanics.)
Using earth as a model again, tropical climates mostly run about 10 degrees north and south. There's a lot of variation here since earth does have Coriolis effects and the placement of land and water has a major impact but we can simplify for our purposes. 10 degrees is about 1/3 the size of earth's Hadley cell so we'll use that to scale our climate zone as well. That works out to about 700km.
Tropical climate within 700km
On earth, we see lots of deserts from the edge of the tropical climate zone out to the end of the Hadley cell. This is, again, complicated by the Coriolis effect and actual land/water placement on earth but we can use it for our model.
Desert climates from 700km to 2200km
So, what lies beyond the desert?
Our "star" is still providing heat out there but it's going to dissipate much faster than on earth because distance will matter more than angle or atmosphere. For example, consider the simple case of 45 degrees. On earth that gets us to the US/Canadian border, the heart of Europe, northern China, or southern Chile/Argentina and parts of New Zealand. These areas aren't known for particularly nasty weather around the equinoxes (which is when we need to model). To the extent that weather is poor, it's typically a matter of geography, not sunlight.
Things are different on this flat world. Light at 45 degrees here must travel 41% further. Ignoring the atmosphere for a moment, there's the inverse square law. Radiation drops off with the square of the distance. If the distance is 1.41x then the insolation is at 50%. Now, add in the extra atmosphere and it drops down to something like 35%. We're getting into proper cold here. Even at the edge of our Hadley cell the effective insolation is down by 10%.
You should get a temperate zone; it'll just be narrow.
Some other factors to consider:
The falling air at the edge of your Hadley cell will impact the air circulation and climate outside zone. As will the flow of air back towards the center. How that will work is much harder to guess given the lack of Coriolis effect.
More importantly, as has been noted in comments, the placement, size, and geography of any land masses will have a major effect on your climate. Some of the above may get completely twisted by mountains, for example.
Finally, what about ocean currents? Depending on how your oceans are laid out, the currents there can also change climate.