When but is used as a preposition like this it means "except". I think that you normally use the bare infinitive after but: - I couldn't do anything but cry. - I could try but fail. There are a few key words that trigger the complete infinitive. "choice", which you picked up on, along with "option", "alternative" and words like that: - I have no choice but to cry. - There's no other option but to succeed. - He had no other alternative but to scream.

It means something like ‘except.’ So you have to use ‘to’ because ‘I don’t have a choice but to’ is an expression itself. In the ‘I can’t help but’ expression there’s no ‘to’ so you don’t use it when making a sentence. You just have to learn that with one you use ‘to’ and with the other you don’t :) Does it make sense?