I am assuming that you chose not to take out a “home warranty”. Otherwise, I don’t think you would be inquiring here as to what your recourse is. Looking to others to cover an appliance failure is an unrealistic expectation.
To your question regarding the 90-day buyback, that program is limited to 90 days after you close. I think it’s safe to say that 161 days is well beyond that.
You have no recourse against the inspector. A home inspection is a snapshot of the condition of a home on the day of the inspection, with no warranties expressed or implied.
The first house my wife and I bought together had a heater that passed inspection, but come the first few cold days, I suspected problems and sure enough, CO levels in the home were over 4 PPM. When the unit was pulled, it looked like someone had thrown a hand grenade in the burner assembly, it was so badly rusted.
This was before I became a Realtor or an inspector, but it would never have occurred to me to go after the inspector. Even then, I knew that for the few hundred bucks we paid for the inspection (back in 98), that it was unrealistic to expect an inspector would have been able to see what was going on deep inside the heater.
As Chuck said, this is home ownership. If you don’t want to pay for a home warranty, save up your money, because things break.