There are instances when I want to use a "but" in a sentence that starts with "but". Is that grammatically correct?
There is nothing wrong with a sentence like that.
But if you were concerned you could break the sentence into constituent parts, and you would see that it was correct, but might still want to consider splitting the sentence in two to make it more readable.
English writers have been doing it for centuries. For example from Edmund Burke, A Philosophical Inquiry into the Origin of Our Ideas of the Sublime and Beautiful (1757):
The swan, confessedly a beautiful bird, has a neck longer than the rest of his body, and but a very short tail : is this a beautiful proportion? we must allow that it is. But then what shall we say to the peacock, who has comparatively but a short neck, with a tail longer than the neck and the rest of the body taken together.
And from M. F. Sadler, The First and Second Epistle to the Corinthians, with Notes Critical and Practical (1889), quoting (I believe) the King James edition of the Bible at First Corinthians, chapter VI, verse 11:
And such were some of you: but ye are washed, but ye are sanctified, but ye are justified in the name of the Lord Jesus, and by the Spirit of our God.
And from John Fox, Of Free Justification by Christ (1583):
If you would have this common to you with Christ, that you can be without sin, what do you leave him, that is peculiar to himself? But if not, what remains then, but that those high mountains of righteousness with which you are so puffed up, should not only fall, but also vanish.