I’m pretty sure if you’ve already activated the trial code on your licenser for Elements, you can’t switch over to Pro (not positive).
There’s no difference between Elements and Pro as far as the install (though Pro comes with a larger Halion companion). Dorico detects what license you have running, and activates E or P accordingly when you run the software.
I’m really overstepping now, since I don’t know your workflow needs, but I always recommend Pro. Once you get used to all the control of Engrave mode and related functionality, you won’t want to go back. And if you crossgrade from Finale or Sibelius, it significantly reduces the cost.
Elements is great for students, or for casual hobbyists (and I’m not referring to skill level [!!], but level of involvement and professional requirements). But for everyone else…