Stay polite and friendly and from a position of "you don't want to deal with not being able to check me in any more than I don't want to go away unchecked-in." I had a gate agent in Toronto who fundamentally misunderstood the wording of a letter permitting me to travel to Singapore.
The gate agent asked "why are you going to Singapore?" and it's not a secret, but she's not an immigration agent trained in determining whether reasons are essential. So I told her "we have an approval letter, it's for an essential reason" and I gave her my son's letter.
That starts
APPLICATION FOR ENTRY IN VIEW OF 2019 NOVEL CORONAVIRUS
[Son's name], Please refer to your application for entry regarding the above matter. I am pleased to inform that you may seek entry to Singapore between 30 Oct 2020 and 03 Nov 2020. The Immigration Officer at the checkpoints will assess your eligibility for entry, provided that you fulfil the usual entry requirements where applicable."
Now, Singapore never says "Congratulations, you are going to be let in." This is permission to present ourselves to an immigration person IN SINGAPORE and say "I would now like to come in." It is our approval letter. But she is all "this is just your application." No, I tell her, the application was 20 pages long, this is the approval. She keeps leaving (with our passports) and coming back and typing and then leaving again. Eventually she comes back with someone else and it occurs to me to show her the PDF from our Arrival Card submission. This is like a landing card used to be, but there are health questions. It is completely unrelated to whether or not you have permission to enter. But it starts:
Dear [Son's name],
You have successfully submitted your arrival information and health declaration to the Immigration & Checkpoints Authority on 30/10/2020 22:55 (Singapore Time).
This proves that we submitted a Covid-related form which was optional, I could have used an app on my phone or waited till we landed, but BECAUSE I DID, I had the magic piece of paper that says "successful" on it.
Several times she said to me "this is your approval. I needed this." And I didn't argue with her because I wanted to get on the plane. And we did.
Looking back now some years later I think she came to realize that she was wrong about the "you may seek entry" not meaning we had permission, and needed a way to save face. She found it on the health form. Had we been argumentative and unpleasant, she might not have gone for it. And definitely bring as many different pieces of paper as you can find, printouts of rules, printouts of emails, receipts, whatever you have, in the hope that one of them might change their mind or give them an excuse to announce "this changes everything" and let you on.