How to get on my PhD supervisor's case without becoming a nuisance?
As confirmed in a comment, the "very senior" supervisor does have a secretary. I suggest explaining the problem, and the reasons for urgency, to the secretary and asking them for help.

The secretary can go to the supervisor's office without an appointment, and is likely to understand that some university paperwork really does have to be done on schedule. Ideally, the secretary will put together a packet with the partially completed report and any other relevant paperwork, such as previous evaluations, that the supervisor has.

I would concentrate on the progress evaluation, because that has to be done, and has already been deferred once.

As far as the travel is concerned, make sure the supervisor, or at least the secretary, understands that you need some notice, and will assume you are not traveling next month unless there is a definite decision that you are traveling.