Consider:
Don't get up – I'll bring a cup of tea to you.
Don't get up – I'll bring a cup of tea for you.
I would think they both can be rephrased into: I'll bring you a cup of tea.
Does the choice of "to" or "for" make a difference here?
Consider:
Don't get up – I'll bring a cup of tea to you.
Don't get up – I'll bring a cup of tea for you.
I would think they both can be rephrased into: I'll bring you a cup of tea.
Does the choice of "to" or "for" make a difference here?
@Amber's comment to the question probably slightly overemphasises one possible distinction, but it's not completely wrong. The first and most important thing to note is that both OP's alternatives are far less likely than the "ditransitive" forms...
1: "Don't get up – I'll bring you a cup of tea"
2: "Don't get up – I'll fetch you a cup of tea"
3: "Don't get up – I'll make you a cup of tea"
If we recast the above into "monotransitive" constructions ("I'll bring/fetch/make a cup of tea for/to you"), we find that to works perfectly well in #1, and "just about" works in #2, but it doesn't work at all with #3. By contrast, for works with all these verbs (and many, many more).
In general, native speakers will (usually, subconsciously) recognise that OP's alternatives aren't the most "natural" phrasing, so they might look for a "reason". And most likely if they do decide on a possible distinction, it'll probably be that bringing a cup of tea to you emphasises to your location, whereas for emphasises for your benefit. But in practice any such distinction normally means little, in context.
To answer the statement that isn't a statement so much as a question in a mask in this context due to its first phrase being 'I think'... yes, both of them can be rephrased into this form, and this is more common and natural. And more pleasant, for the blessed receiver of said cup of tea.
The difference between the 'for' and the 'to' here is (as far as I'm aware) the verb. Let's try replacing bring with give:
Don't get up – I'll give a cup of tea to you.
Don't get up – I'll give a cup of tea for you.
You should notice here that the second one is definitely not natural to say. This is simply because you give something to a person.
However, when you bring something, you bring it for them, because it is considered a gift in this context.
When you talk about bringing a cup of tea, the cup of tea is a gift and is thus brought for them.
...Final word: using 'for' is certainly more natural than using 'to'; though 'to' is correct I wouldn't recommend using it instead of 'for'. Then again, I'd recommend you rephrase the sentence to: "I'll bring you a cup of tea", as you mentioned.
If there is any difference, it's subtle. Your third formulation (I'll bring you a cup of tea.) is certainly the best.
I might say the first if I wanted to emphasize that they need not come to get the tea -- the tea is coming to them.
And I might say the second if I wanted to emphasize that among many cups of team I'm bringing, one will be for them.