Wish you lived in OZ.
Think of the tradition "swearing by the bible". It's just something the legal profession conform to even if it makes no sense and probably has no effect on truthfulness when querying witnesses.
You can still have your own beliefs. BTW I really like that US series 'The Hard Fight'. You may not get it there as it's very political. I watch online.
We all compromise, that's part of the game.
And it is a game. That's the way to look at it when sometimes things don't make logical sense.

Is this criminal or civil? I was a prosecutor for two years and did several jury trials. As the prosecutor I’d be ecstatic to ask the judge for a modified oath. I just wouldn’t want that in front of the jury. Especially not in a place like ND. So he’d modify the oath with you and maybe an attentive juror would pick up on it but they’d never have a way to confirm it. Also if you let me know enough in advance I could ask questions during jury selection to kick the ones that might hold it against you and question your credibility because of it.

All you need to do is ask the attorney who is calling you as a witness to ask the judge in the pretrial conference to allow you a secular alternative. This is an issue of capacity to testify. (As a side note, infuriatingly, atheists used to fall in the same category as very young children or insane!) Every state should allow a “solemn affirmation” or something like it instead of swearing on a bible. The federal rules and ND say that there doesn’t need to be a specific oath as long as, to the judge’s satisfaction, the individual understands the difference between lying and telling the truth and understands he or she is under oath.

Actually there are accepted alternatives that you can use in place of god . We use them in our wedding ceremony and at several other legal formats. One is by "my conscience" another is
"by my truth in statement"

Since the bible has no more significance to me than a stack of comic books, it's just as well to swear on it than anything else. Hard to think of an object I could put my hand on that would have meaning in that context. "Do you swear on this copy of Springsteen's "Born to Run album?" I'd let it go.

I'd recommend just going with the flow. It's a "tradition" in our judicial system. While you're there as a witness is probably not the best time to try changing stuff..... Just my opinion, of course.

I'm with @icolan on this. If you support and agree with the justice system and trial by jury, then you probably want it to have integrity. There's only so much we can do in order to maintain integrity in the system. As a witness, one can do their small part by being honest in their affirmation and their testimony.

Hmm... Just spitballing here, but I would think a judge would (should, at least) be educated and established in impartiality enough to know that atheists > logic & reason > objectivity > truthfulness, trustworthiness. By those lights I would think that atheism can be a boost to one's credibility in court. Justice is blind, no?

Most courts no longer require that someone swear to god, on a bible.
You still have the option to decline, if you wish to.
As far as being concerned about how others will view you, or your testimony, if you
refuse to swear to god, that's something only you can determine as being important or not. Personally, my own integrity is more important to me than what anyone else is going to think of me. If I lie, and violate my own principles, it's likely that I'll be the only one to know. Can you live with lying? That's the only question that really matters.

Why cause trouble, do it for their peace of mind. If they get a false sense of security from swearing the truth over a book of lies, who are you to argue with.

The jury enters the courtroom After the Judge does, and After any swearing in, it will never be revealed to them!