Need to verify what is in your breaker box.
usually they are 220 volt, the breakers on the left are one leg at 110 volt, the breakers on the right are a different leg of 110 volt.
So is the plug on your breaker box wired to one leg? Or is it wired into both legs?
if it is wired into one leg, then it should work. If wired into both legs then that is your problem.
When I was running 110, I moved all my "needed" circuits to one side of the box.
Feeding now with 220, I just turn on whatever I need.
I would probably be feeding from the 220 circuit on your Gen, probably solve your problem.
Most likely your furnace and well pump are both 220 volt.
For me, I have a 5k generator. Once I get it up and running I have to start my furnace first. Because startup takes the most current.
After it is up and running, I switch the fan to ON so it stays on. No more startup spikes.
Then I will turn on circuits I need.
When i had a well I would have to alternate between the well pump and the furnace, 5k would not run both all the time, usually when the well kicked on it would just trip out.
I would heat the house up, then swap to well water so everyone could shower and we could cook, then back to heat for the night.
That would be the main right