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.
