Temperature change is caused by lapse rate
The difference in temperature between the bottom and top of the crater is going to be driven by the lapse rate, and not by the heat of the crust (probably). Lapse rate is the rate at which temperature in Earth's atmosphere decreases with an increase in altitude, or increases with the decrease in altitude. In particular, we can use a chart of moist adiabatic lapse rate to see what the expected temperature changes would be in a 5 mile (8 km) deep crater.
The 'Moist Adiabat's are the dotted lines on the chart. Chicago has a annual mean temp of about 10 C, Washington DC about 15 C, Houston about 20 C, and Miami 25 C. Pick your starting point, then follow the dotted line upwards until it intersects with the line representing 8 km. I get the following:
Floor Temp Surface Temp Surface Press
10 C -60 C 0.34 atm
15 C -45 C 0.36 atm
20 C -30 C 0.38 atm
25 C -15 C 0.42 atm
So you are looking at quite the different in temps. I also used this chart to double check your pressure estimates. Instead of starting at 1 atm, your crater surface starts at 1.2 atm, so I multiplied the resulting pressure estimates by 1.2. You are pretty close to being right on the pressure.
Overall, the area outside the crater will be very uninhabitable.
Why the Earth won't provide appreciable heating in the crater
The Earth's crust is about 30-50 km deep on the continents, away from tectonic boundaries, and it has a temperature gradient of about 200-400 C in heating between the surface and the boundary with the mantle. The heat flux on continental crust is about 71 mW/m$^2$.
The simplest argument I can make is to compare the heat flux on the continental crust with the heat emitted by the Earth's surface in infra-red radiation. The Earth's surface emits 398 W/m$^2$ in IR radiation to the skies, on average. This is significantly higher than the 71 mW/m$^2$ that comes from inside the Earth.
Lets look at what happens to planetary heat flux when we change the thickness of the crust. Planetary heat flux is controlled by the Fourier's law for head conduction
$$q = -k\nabla T.$$
Here $q$ is heat flux (W/m$^2$), $k$ is conductivity and $\nabla T$ is the heat gradient. If we reduce the distance between crust and surface by 8 km, from 25 km to 17 km, with a constant 300 K temperature differential then $\nabla T$ goes from 12 K/km to 17.6 K/km. This represents a 1.5 times increase in heat flux out of the Earth's surface.
So the Earth is providing something like 120 mW/m$^2$ at the bottom of your crater instead of 70 mW/m$^2$. Still, compared to 160 W/m$^2$ absorbed from sunlight and 80 W/m$^2$ released by evapo-transpiration and IR radiation to space, and back-radiation received from atmosphere and clouds....you get the picture. The change in planetary heat flux is negligible, three orders of magnitude smaller, at least.
Therefore, the temperature gradient between the crater bottom and high surface are going to be driven mostly by lapse rate, the same factor that drives the temperature difference between sea level and the top of Mount Everest on Earth.
Air flow around the crater
Your crater is much hotter than the surrounding air. Hot air tends to rise. The dominant climactic feature will be rising hot air out of the crater. This will cause a variety of follow on effects.
Rising hot air creates cyclones!
Your crater is so large it will induce high speed, cyclonic winds circling around it. In the diagram above, the 'x' and 'o' represent winds into and out of the page around the crater. There will be permanent winds swirling around the crater.
If your crater is entirely in the northern or southern hemisphere (assuming your planet is rotating like Earth) the Coriolis effect will drive the winds into a stable clockwise or anti-clockwise rotation around the crater. If the crater straddles the equator....something will happen, I'm not really sure. if the crater is on the Equator and large enough relative to the size of the planet, the cyclone may actually form a circle around the surface of the planet, but don't quote me on that.
Rising air releases moisture as it cools
As your air rises, it will lose its ability to hold moisture. You can see this from the thermodynamic diagram at the top of the page. If we assume a jungle-y 25 C average temperatures at the bottom of the crater, then that air can hold about 20 g of water per cubic meter. Elevate that air to 8 km, and it can hold about 3.5 g, leaving the remainder to fall as rain.
The center of your crater will be constantly raining. The temperature gradient between crater bottom and surface is going to be much higher than the temperature gradient between day and night, therefore, you will always have a steadily rising column of hot air.
Various wind conditions might blow around pockets of warm air, especially towards the edges, but you can assume the center of the crater will see rain every single day. The hotter the crater, the higher the magnitude difference in saturation mixing ratio will be, so the more rain you will get. A 10 C crater will get light rain every day, a 25 C crater will see permanent torrential rainfall.
Descending cold air will enter the crater like a blast furnace
From the cyclone swirl, descending cold air will spiral into the basin to take the place of the air that rose out. The force of gravity will ram this wind into higher pressure areas at the bottom of the crater, and the resulting molecular friction will be expressed as adiabatic heating.
These are foehn winds. The 8 km drop means that you would see an expected 30-60 C of adiabatic heating as the wind rushes downhill. Given that the wind already has high kinetic energy from the hurricane swirl at the top, the result near the bottom with be superheated blast-furnace winds. If conditions are right at the top, with extra solar heating for whatever reason, you could easily see steady 45 C or higher gale-force winds at the bottom of the crater.
No trees will live on the slopes of the crater, due to the high wind speeds and highly variable temperatures. The slopes could easily see temperature changes of 30 C in a matter of minutes, as one air mass blast past and is replace by another.