Robert, you are not understanding the difference between prepared and learned.
You do not prepare your cantrips; You choose which ones you learn at 1st level. You cannot CHANGE them; you just know them. At higher levels you will be able to learn more cantrips, up to 6 (in normal rules).
You prepare spells of 1st level or higher, and you can change them each LR (or in the game, at any time out of combat). You cannot, however, change your cantrips in this manner because they are not prepared.
In the game right now, you can scribe cantrips to learn new ones - this shouldn't be happening, but either way, if you do scribe a new one, again, it's just something you now know, without limitation. You don't have to choose between which cantrips you do or don't have access to on a particular day, because you always have access to every cantrip you know, every day, without preparing them.
I would encourage you to roll a wizard, as you say, and then get to a point where you can rest. Try to change your cantrips to four different ones that you had the option of picking on the character creation screen - you'll see that you cannot.
In more detail, to speak about BG3 specifically:
In character creation, you'll note that there are separate headings for cantrip, then spells, then, below that, prepared spells.
The first section lists the game's wizard cantrips for you - as you noted, there's 12 of them. You can pick 3 at character creation.
The second section presents all 1st level wizard spells - these are different from cantrips, which are functionally 'level 0' spells. You can pick 6 of these to learn at 1st level.
The third section asks you which of your spells you want to prepare. Cantrips are not presented here; you don't prepare them, you just know the ones you picked. This list presents you the 6 1st level spells you just selected above, asking you to choose which 4 of those you want prepared when you start the game.
Once you begin the game, you have the freedom to change your spells at any time when you're not in combat. You can open your spell book by pressing 'k' by default.
Here, you'll see your known spells, arrayed by level. Your 3 cantrips will be at the top, on a line of their own. You might have an additional one if you picked a race that gets a bonus cantrip. Notice how they are lit up, but do not have a selection line around them. This means they're available to use, but aren't counting against your prepared spells limit.
Below that, you'll have your first level spells. There will be 6 of them, and four of them, the four you selected in character creation, will be lit up, and will have white border around them; this means that they are available to cast and are counting against your prepared spells.
At the bottom of the spell book ,you'll see a blue bar that represents you prepared spells - it's showing you that you can prepare up to 4 spells, and that you currently have 4 prepared. If you click one of the spells with the white border, it will unpreapre that spell, letting you select a different one.
