No
The Spellcasting feature does not allow for changing of cantrips, by the same justifications as outlined in the answers to the questions linked. Briefly:
- "retraining" spells is explained under the heading "Spells known of level 1 or higher", clearly excluding cantrips
- The rule specifies that the new spell must be of a level the caster has slots of (which can never be true for cantrips)
- In-universe reasoning: Cantrips are learned through long-term rote practice, "fixing them" in the caster's mind
So there is a rule in the books specifically stating that cantrips can be replaced.
The rule doesn't actually say that --- it's more along the lines of "if a cantrip could be replaced, this is what would happen". However, it should be noted that the statement refers to both cantrips and spells of level 1 or higher. The latter can be replaced as usual.
You don't "gain" your first character level
The rules for creating characters define what "gaining a level" means in the beyond first level section:
Beyond 1st Level
As your character goes on adventures and overcomes challenges, he or she gains experience, represented by experience points. A character who reaches a specified experience point total advances in capability. This advancement is called gaining a level.
The entire section repeatedly uses the wording "when you gain a level" for things that you obtain on levels beyond the first.
An example of another class that gets features when they "gain a level" is the wizard's spellcasting section which states:
Learning Spells of 1st Level and Higher
Each time you gain a wizard level, you can add two wizard spells of your choice to your spellbook for free.
If you also considered that you "gained" your first wizard level, you would then add an additional 2 spells to your spellbook for free beyond the 6 your spellbook starts with. I have never seen anyone consider that this is the case and the pregen wizard character only starts with 6 1st level spells which supports the interpretation that the first character level is not really "gained".
Therefore, the easier interpretation is that your features that state "when you gain a level" in general should apply to getting levels beyond first.
Multiclassing is difficult
You also stated that you were interested in knowing if multiclassing would change this. Indeed, the rules for multiclassing do state that you are "gaining a first level in a class" in the section introduction:
With this rule, you have the option of gaining a level in a new class whenever you advance in level, instead of gaining a level in your current class.
And when mentioning gained proficiencies:
When you gain your first level in a class other than your initial class, you gain only some of new class's starting proficiencies, as shown in the Multiclassing Proficiencies table.
It is also no longer your first character level so the reasoning used above would not apply. This would allow the interpretation that you are indeed "gaining a sorcerer level", therefore letting you retrain the spell as you wanted.
That said, though, considering multiclassing is already an optional rule that you need explicit DM permission to use, asking your DM about whether multiclassing changes this for a Divine Soul Sorcerer would be part of the conversation to have with your DM.
Personal Opinion
As a DM, I'd not allow multiclassing to change whether you can retrain the first level spell. If a player really wanted to get the extra retraining on sorcerer level 1, I might just allow it for a single classed character (though note how that makes the whole thing about choosing the alignment completely pointless, you might as well just gain any 1st level cleric spell) or just not allow it at all even with multiclassing. (Similarly, I'd also not have a multiclassed wizard start with 8 spells, that's just weird to me).
Best Answer
Yes.
Divine Magic doesn’t restrict the sorcerer to the cleric spell list in the Player’s Handbook, it just says that they can use the cleric spell list. Tasha’s Cauldron of Everything is explicitly adding to that list—so divine soul sorcerers benefit just the same as clerics do.