Definitely do so standing up, because just in the process of lying down there's a good chance it might get pressed out and leak. I'd recommend removing all of the clothes from your lower body, and if your shirt hangs down lower your shirt. Change on a surface that you can clean easily as there's a chance some might leak no matter how hard you try. When you lower the diaper, depending on how well the feces are contained, you may be able to get away with sliding it down after only having the top tapers undone, this will help the diaper keep its shape and pocket per se, to help contain the feces instead of straightening out (with high leak guards this may not be too big an issue), but be careful depending on the fit that you're not smearing feces down your legs in which case you'll need to undo both tapes and lower it down carefully. As you're lowering it also try to wipe as much as you can off your body using the back of the diaper to avoid it dripping. You can even have a few wipes ready and have them at the back of your diaper as you lower it to try to start cleaning off some of the mess.
If necessary you can also try using some non-prescription medication for diarrhea relief if possible to control it to try to relieve it at least partially or put it off until the end of the later (for example using imodium before outings) as even small fixes help