You haven't specified the type of star, but to get a tidally locked planet within the habitable zone it will have to be an M (red dwarf) or K (orange dwarf), at a distance of ≤0.6 AU:
This also means the planet's period will be somewhere between 12 and 219 days, a little short of the desired 250, according to some quick calculations (try it here, keeping in mind you need to use values to the left of the Tidal lock radius that fall within the yellow Habitable zone). This means your Coriolis effect will be stronger than anticipated, but that's a good thing because you want the atmosphere to circulate.
Rather than a concentric ring of superheated air, newer models predict the Coriolis force will 'stretch' the substellar atmosphere longitudinally, as in this example:
Source: https://www.pnas.org/content/111/2/629
The left column (A & B) shows direction and speed of surface winds, the right (C & D) ocean currents. Keep in mind this is a model of different ocean planets, with white as sea ice and blue as open ocean, but the pattern holds across multiple simulations: each hemisphere its own atmospheric cell, with temperature, pressure, and so forth, stretching and mirrored along the equator:
Source: https://www.nas.nasa.gov/SC19/demos/demo30.html
Source: http://serious-science.org/the-climate-and-habitability-of-planets-with-eternal-day-and-night-sides-5289
You'll notice the orbital period and other details vary from one simulation to the next, with a general trend of less 'stretch' the longer the period is, with all periods so far being in the low to mid double digits.
To me this suggests climates would be mirrored along the equator for most of the substellar face, with the coldest climates forming arcs across the antistellar face above and below the warmer (but barren) nightside equator.
If you're wed to the 200+ day period and its ultra-low Coriolis, here's what seems to happen when the period approaches 100 days (far right):
Source: https://www.researchgate.net/figure/Different-troposphere-circulation-states-identified-by-Carone-et-al-2015-for-tidally_fig1_321418244
Radial flow, or a single cell filling most of the substellar face of the planet, with what appears to be an atmospheric ridge just this side of the terminator. (Keep in mind this diagram shows the upper atmosphere, so surface winds would be the opposite direction).
From this I would assume a tropical climate at the substellar point, drying to savannah and then hot deserts as it approaches the Terminator, a ring of temperate trees grabbing the last of the useable sunlight at that ridge, then a rapid transition to colder grasslands and shrub-covered sun-facing hillsides before the dark wastes beyond.
The longer the period gets, the smaller the tropical zone and larger the hot desert would get.
Without knowing where your continents would be and how they'd shape ocean currents, and how those currents would interact with the prevailing winds, I don't think it's possible to get more specific than this. 188