A short bed truck is fine, it will just limit your options a bit.
Depending on how many people will go on your trips, it may be a tight squeeze, but if you only use the camper for sleeping in and keeping warm, you should be fine.
If you want to winter camp, you want enclosed and heated tanks in the camper, not exposed tanks that hang off the back end (like mine in my sig).
A four season camper for a short bed truck might be a little harder to find, depending on your budget and location. Location is important. I had to drive 1000 miles to pick up my camper, because it was a short bed camper and the time of the year impacted availability of used units.
Hardside campers for winter, look for Northern Lite and Bigfoot, mainly. Some Lance meet the criteria, older S&S campers, Arctic Fox or Eagle camp.
The problem with short bed trucks is that you have a center of gravity issue, so the camper has to be designed right for short bed trucks, and you will have limited payload capability.
Pay close attention to your tire weight limits, and you will be fine.
2016 F250 CCSB 4x4 6.2L
2001 Lance 820