I'm not a fan of most mesh WiFi solutions that seem to be popular lately. It is much better to run an Ethernet line and add a second wifi access point (the key here is wired back haul). I understand that for a lot of people the mesh may be cheaper and less work, and they may have restrictions. But you end up with an inferior network with a back haul bottleneck (in most cases.)
To alleviate the back haul bottleneck Eero has tried to do something they probably think is clever but has some compatibility issues with some older or cheaper IoT devices. Unfortunately the Ambient Weather WS-2902A is one of those devices. Not only does the the WS-2902A work on the 2.4 GHz band only but I suspect it only works when the WiFi router is using 20 MHz channel widths (the standard width for a channel). But when the Eero sets up its mesh in order to reduce the backhaul bottleneck it bumps up the channel width to 40 MHz on the 2.4 GHz band. This is a known problem for a lot of devices that will only work with the standard channel width of 20 MHz. Sonos and older Chromecast come to mind with this problem (haven't checked lately but maybe this has been resolved with newer firmware on those devices.) This wider channel use is not a problem for modern laptops and mobile phones and other such devices with better WiFi technology.
There is a trick to fix the Eero to not use the wide 40 MHz channels. You have to set your Eero to only use channel 11 on the 2.4 band. Since channel 11 is the last channel available it doesn't have room to grow to 40 MHz. The Eero probably defaults to automatic channel setting and this means it will use channels 1 through 10 in wide mode. But wide mode is basically the same thing as using more than one channel at a time. So channel 1 wide at 40 MHz is like using channels 1 and 2 at the same time. By forcing the use of channel 11 you have limited the ceiling to 20 MHz because you can't use channel 12 in the USA as it does not exist per FCC regulations. Each channel is only 20 MHz. A 40 MHz channel is really just two adjoining channels.
But before you just jump on forcing your Eero to only use channel 11, reconsider a better WiFi solution. Because using a 20 MHz wide backhaul on a mesh network is going to more than half your WiFi performance (it was already not great using a mesh to begin with and you are about to make it worse.) Basically you are taking the "mesh" out of Eero and making it like a cheap WiFi Extender. As I first mentioned maybe taking the time and trouble and expense of a wired backhaul for a second WiFi access point may be worth considering.
Another thing you can do is consider a more expensive WiFi main router like an Asus 5300. That thing is an animal and puts out a crazy strong signal. I've installed these in a few troublesome locations an I've not needed a second access point. It's a costly router but in a congested area or on a large house up to 5,000 sqft it performs. Another thing to consider is a mesh network by Asus (get a couple RT-AC68U and configure for mesh) as they do the backhaul via 5 GHz and dont mess up the 2.4 GHz band with 40 MHz channels. We'll Asus does support mixed 20 MHz and 40 MHz channels but it switches it on and off automatically depending on what the WiFi client device needs (it's more backward compatible). You can also do a mesh with the pair of Asus 5300 or one 5300 and one 68U or other models that support Asus mesh (3 or more can mesh also). Another thing I like about the Asus mesh is that it can also support a wired backhaul, so you can start with Asus on WiFi mesh backhaul and then cable up and improve performance. If you get a pair of 5300 they can do the backhaul on the 3rd radio band as they have two separate 5 GHz radios. In this case you don't have to sacrifice the 5 GHz band to be the backhaul because you have an extra 5 GHz band. I'm a big Asus fan as you can tell. But I still prefer regular Ethernet cabled WiFi access points.
Realize that the difference between a WiFi mesh network and a simple WiFi extender is this added capacity for the backhaul (use of wider channels for backhaul or 5 GHz backhaul or Ethernet backhaul). Years ago before mesh existed, the cheap WiFi range increasing solution was an Extender. The Extender or Range Booster all it did was repeat the signal and half the speed because of the shared backhaul with the signal itself. So if you remove your Eero's mesh backhaul enhancement for the sake of your weather station, then you basically overpaid for what you could have accomplished with a cheaper simple WiFi extender that wasn't "mesh." But better to have ventured and learned. Now you know why not all mesh are equal, and why Ethernet backhaul is better and may be worth the trouble (I realize some people rent or whatever else have cabling restrictions, but for those people I say get a better mesh than Eero.)
I'm not for certain saying this is your issue. I'm just sharing what I know is a problem with eero and other mesh type networks. Test it out and let me know what you find out. Ambient Weather on their instructions tell you to turn off your mesh network by unplugging other nodes except the main. They don't explain why. But I suspect the reason is for the explanation I've detailed.
Another option for you is to just use the WS-2902A display console as just a display and not connect it to the Internet. Then you can add the ObserverIP (and a WS-1000-BTH) and then that is connected to Ethernet and that does the reporting to ambientweather.net and Weather Underground. You will then lose WeatherCloud. But you can now upload to PWSweather.com via ambientweather.net. But then you can take it further and connect a Meteobridge (or WeatherBridge) to the ObserverIP and then send to WeatherCloud and many more places including your own website or SQL database...endless options with a Meteobridge. Or connect a Raspberry Pi to the ObserverIP and run WeeWx and do similar things. So not having the console connected to the Internet is not a must have solution if you do it via ObserverIP.
Or yet another solution may be to try the WS-2000 display console (buy just display separately - not sure if available yet) as that may not have the compatibility problems of the WS-2902A display console in regards to WiFi. You'll then need its optional separate indoor temp/hum/pressure sensor.