This is not how it works. The AP is not actually responding to or reflecting the radar ping so the range detection won't work. If it even sees it it'll show up as a line from the center of the radar screen in the direction of the AP. It won't pinpoint it. But most likely it'll just add to the noise floor.
It'll still be possible to locate it with that info but it's more work unless you manage to interfere with two radars and they can triangulate. That's pretty unlikely for an indoor access point with only a few milliwatts though.
I saw meteorologists publicly shame rogue APs by name. This implies they do have means (directional antenna) to probe the source of interference and get information such as AP name, MAC address and distance.
They wouldn't get this from the actual radar though. It must have been the result of further investigation.
I was mainly referring to the 'radars are good at pinpointing things' comment from the OP :) I wanted to clarify that an AP on this frequency won't show up as a dot on a radar screen. For the distance to be captured it must have an actual reflection.
You can even see some of the lines I mentioned in that pic!
Radar receivers have a secondary output (or at least a waveguide coupler) where you can connect arbitrary equipment. So you connect standard wifi card to it, enable monitor mode and listen to all the MAC addresses and SSIDs (if they are not hidden) - directly though the giant radar antenna. (I really need to try it with our radar, I currently only detect packets based on their wide spectrum, but I have never decoded them)
There is even a software-based wifi demodulator (https://www.nuand.com/bladerf-wiphy/) so you can try to decode it from the IQ data, but they usually don't have enough bandwidth.
The way this works is someone notices a disruption and notifies the relevant authorities, which have detector vans to track down the emission sources. And they're quite good at that, too.
The comment mentions "something like a single suburban home". I suspect a line drawn on the map in the direction of the signal may give a good enough idea which few suburban homes may be the culprits, simplifying the task of locating it from a van.
I may overestimate the angular precision of that though.
The beamwidth of a standard weather radar is 1 degree. You can get to maybe 0.3° accuracy if you are pinpointing the maximum carefully. So this is 400m at 70km distance - which is not that bad. You then need to evaluate which parts of this sector are actually visible from your radar site (using a digital elevation model) and then search in them (I have never done this). Depending on the location, the entire 100km line may be visible...
But you need to specially do this scan and carefully evaluate the maximum (tricky if it's transmitting only once in a while). When you are scanning normally, you are pointing a bit above the ground (you are scanning the sky, of course) and the beam may have several degrees at the ground level.
It'll still be possible to locate it with that info but it's more work unless you manage to interfere with two radars and they can triangulate. That's pretty unlikely for an indoor access point with only a few milliwatts though.