Tire carrier with receiver hitch
Our Greyhawk also lacks a spare. I haven't decided yet if I am going to add one and if so, where to carry it. The rear cargo bin is one possibility.
I am not sure if an under-frame carrier would work. Everything hangs so darned low as it is.
But I did find a Roadmaster hitch-mounted carrier with a pass-thru receiver that would still allow me to hang the bicycle rack on it or even pull my motorcycle trailer. The manufacture says it is rated for up to 400 pounds of tongue weight which is far less than my Kendon trailer with the Honda Shadow on it.
2014 Greyhawk 31FK
2007 Honda Shadow Sabre 1100cc