I totally respect your wanting to have your story together before you go, mate. Good on you!
Something as clear cut as a condom breaking would have to suggest that you know your date of conception regardless??? Surely?
I'd definitely be explaining that you'd trust your dates over a late ultrasound.
And if they pull out the 'you are placing your baby in danger' attitude, say you'll be monitored daily to make sure baby/fluid levels are ok but you'll not be induced.
Perhaps try to find some articles from respected sources that show how being induced can increase the chance of further intervention, etc, and therefore more chance of bub needing assistance as well? You can use that argument to refute the 'you are putting your baby in danger' line if it happens to be pulled out. I think there are some summaries in New Active Birth that I can look up for you if you like. Or there may be some references on articles on this site - I haven't looked around recently.
But hopefully you WILL be pleasantly surprised and your OB will be a darling and bow out, admitting that you are just fine and don't need to even think of induction at this early stage.