Well Hugh , you do have a problem.
I own a 1995 F-350 ,4x4 .
I run the 12000 lb Reece receiver hitch and have no problems mounting my spare under the rig where it belongs.
I think you should re think your hitch.
Based on your comment, I revisited installing the spare under the bed. I had all brand new hardware to do it but couldn't get it to fit last time. So, I took a look at things again and found 2 holes that would work for hanging the spare. I'm not sure they are the right holes but they are holes that work. I think what was throwing me off last time was the hanger for the spare on the driver's side. There is clearly a bracket mounted to the frame that looks like where the hanger would mount but THAT hole would not work. I found a hole on the frame further towards the axle and THAT ONE worked. Not every having the hardware meant that I was unsure which holes to use. There's is plenty of clearance around the spare and it doesn't touch anything so I was able to mount it under the bed. Looks good and frees up my front hitch for a bike rack or winch.
Thanks for making me think more about this!
Hugh