Wireless thermostat issue. An odd problem for sure

Hey guys, this is the second time typing this so I hope it works. I am having an odd thermostat/furnace and am hoping you can help.

I have a house from 86 with what i believe is original gas furnaces. One large furnace for basement/main floor and one for upstairs where the bedrooms are.

In the summer we got central air installed in the smaller furnace because during the summer our bedrooms are saunas. The house was wired with 2 wire thermostats and new wires were next to impossible to run so they opted for a wireless thermostat system. The two wire thermostat was taken down and the wireless thermostat put in it's place.

The wireless thermostat worked perfectly for the central air/ Fast forward to the winter, we try to turn the furnace in question on, but the burner does not ignite. The pilot light is on, and the fan can be turned on and off at the on/off switch hooked up to the furnace. For some odd reason the wireless thermostat would not let the furnace ignite the burner. We have gone for a while without this furnace as it is not a huge issue because the main furnace is large enough to heat the house, but it get a bit nippy at night. My gas provider came and gas is not the issue.

Fast forward to yesterday when I feel irritated that the furnace worked perfectly last year, but coincidentally stopped working when I got the wireless thermostat and central air installed. i mean I could see if the furnace worked for a while and then decided not to, but it just didnt work since we first turned it on this winter. I hooked up the wired thermostat and the burner turns on and works perfectly. the odd thing is that the wireless thermostat must be on the same setting as well in order for it to work. I find it strange that the wireless thermostat did diddly squat with the furnace prior to me installing the wired thermostat, but BOTH are now required to function in order for it to work now.

I was wondering how I can disable/unplug this wireless thermostat for the winter so i can use just the wired one, and then I will switch it back in the summer. Having two makes it near impossible to make a schedule work properly, and it is pretty dumb I need two thermostats.

I have attached some pictures of the wireless thermostat (Honeywell), wired thermostat (Vivint) as well as other related bits.