"Your dog is so cute!"
"Your dog is so cute" is a perfectly appropriate translation, as the word 太 as part of the structure 太...了 typically means "so" rather than "too." You're right that 太 can mean "too," but it has several other meanings as well, and the interpretation depends on the context.
Conversely, in 你的狗很可爱, my understanding is that the word 很 would simply mean "is" rather than "very," since it's the default word connecting a noun to its adjective in descriptions.
很 in this context (connecting a noun to an adjective describing it) simply means "is," according to DL's grammar notes. The construct 太...了, while unfortunately not explained yet in the grammar notes, seems to convey the emphasis on degree that's found in the original English sentence, which would be lost if 很 was used.
太 often means "too [much]," but it's also used in this sort of context in a way that's conveniently analogous to the colloquial English use of "too" to mean "so" or "very"; think, "your dog is too cute!"