What are one's options when facing religious discrimination at the airport?

I fail to see where in this case, quoted in the question, that it is a clear case of discrimination.

Two other Air Canada employees then approached Fatima, reiterating she must remove the head covering because she wasn’t wearing one in her passport photo.

From the text, I assume, we are talking about a US Citizen with a US Passport.

Passport Photos

You cannot wear a hat or head covering.

• If you wear a hat or head covering for religious purposes, submit a signed statement that verifies that the hat or head covering in your photo is part of traditional religious attire worn continuously in public.

If the person in question had no religious problem when the passport was issued, they should have no problem when using the passport.

In cases when the passport is issued with a headscarf that can seen in the photo, then it would be a case of discrimination if the person was required to remove it.

To prevent this happening with you wife, insure that her passport is issued with a photo that conforms to her religious beliefs.

The best thing you can do is to carefully study the rules & regulations that unfortunately are different for every country & airline.

As long as you are operating within the stated rules you and your family should be ok. Millions of Muslims fly every day without any sort trouble or incidents.

If you feel that any specific rule or regulation is unacceptable to you, than you may have to consider choosing a different airline, destination or not fly at all.

Most airlines and countries will make reasonable accommodations for religious needs but there has to be a limit and that's NOT discriminatory. Consider the case of the Kirpan, which is a a ceremonial dagger that many Sikhs feel required to carry. Some countries and airlines do accommodations for this but it varies. There is no desire or intent to discriminate against Sikhs but there also needs to be a limit on what weapons can be on board, otherwise the whole security process would be pointless.

Note: I refer to the TSA in this answer, but it applies to pretty much anyone (who has some authority over you) that you deal with at the airport.

1. What are my rights as a passenger to prevent such discrimination?
2. How can I prevent airline employees from insulting me and my family in front of everyone at the airport?

Answering your question exactly as it is phrased; there is no "right to prevent discrimination", and you cannot stop anyone from issuing an insult without physically preventing them to speak.

The thing is, there's little you can do to stop it when it happens; but you can retroactively file a complaint or seek recompensation if you suffer undue consequences. To that extent, your question of how to stop it before/as it happens can only be answered with "you can't", at least not without severe consequences (such as making a bigger transgression yourself, being detained, or simply denied to board your flight).

I do understand where you're coming from. But the problem is that I can't tell you where to draw the line on taking a stand.

If you feel like you're being subjected to unfair treatment, but the TSA agent is adamant about their (let's call it misguided) position; do you want to stand on principle and risk not boarding your flight, or would you rather ensure that you can board your flight? I can't answer that for you. There are financial and moral ramification to either option, and you need to weigh your own priorities here.

If you want to avoid escalation while still attempting to avoid unfair treatment, you can ask for a supervisor to oversee/confirm what the TSA tells you.
But I've also heard about cases where this response actually caused an escalation, or where the supervisor shares the same (misguided) position and thus doesn't resolve the situation for you.

You can study up on the related rules and regulations, you could even take a printed version of it with you; but I can't guarantee that the TSA agent (or their supervisor) are going to not believe you at your word, or interpret your response as combative.

There is no universal surefire way to stop a TSA agent (or anyone else for that matter) from making a mistake. In the end, at the moment of dealing with them, they have the final say on whether you are allowed to board your flight. If they are wrong and you are unable to convince them of that in the moment, your only other option is to retroactively seek recompensation for unfair treatment.

Depending on what exactly your wife is wearing, you may consider avoiding travel to countries which tend to value secularism over religion. For instance, in France your wife can be fined for wearing a full-face veil in public areas, including airports. She can also be requested to show her hair for identification purposes, although if you insist, the check will be done in a private area by female personnel. The latter will obviously result in a significant delay, so be sure to arrive to the airport well in advance.

You fail to see the other side of this exchange.

The security checks at the airport serve multiple purposes, and one of them is to ensure that the person entering the airport is a registered passenger, i.e. is the person the boarding pass has been issued to. For the purpose of identification, a comparison between the passport picture and the actual person may be necessary.

The interest of dozens or hundreds of other passengers are on the other side of the equation, which is why the airport authorities have rights that can infringe upon certain individual rights of less value. Not feeling comfortable about showing ones face or hair in public, for religious or personal reasons, is a lesser right in this case than multiple people's fear for their safety.

There is no discrimination, because every passenger is verified at this step, it just happens that some make it more difficult. Many men experience scrutiny if their choice of facial hair has changed between now and their passport picture, for example. There is also no insult intended, because the purpose of removing a scarf is not to hurt you or your family, but to complete a mandatory step of the security procedure.

If you understand that, you can find and offer solutions that will satisfy both sides. Your wife can remove her scarf in a closed room with only female security officials present, for example. They get their wish to positively identify her and she gets her wish to not show herself in public.

If you consider that an insult, you might want to consider not flying at all. I'm a frequent flyer and a white male, and while I'm commonly waved through, I have had my belt, my pockets, my carry-on bag and my shoes examined at various occasions. I've had full pat-downs as well. It's a part of airport protocol.