The difference is the use of the verb 'do,' in sentences four and five. The bare infinitive is used after 'do.' So the rule applies because of this, not because of except.

Good question. I had to think long and hard about the answer, and here is the best answer I can come up with (which may be wrong). I think it has to do with what words are implied but omitted after EXCEPT. Examples: I rarely need to go into the city center except [when I need] to do some shopping. She had no choice except [she could choose] to wait for the next train. I drove all the way without stopping except [stopping] to buy petrol. There is nothing more the doctor can do except [she can] keep an eye on him. I have nothing to do except [I can] sit by the pool and relax.

Sentence 2: When we use the phrase "have no choice but to do something. " we usually use to-infinitive. " So, except is used the same way. "He had no choice but to quit." "I had no choice other than to accept what he said." "Many mothers have little choice but to work."