Is it wise to pretend to be Christian to safely travel through the US?

As @MarkMayo pointed out there is no official religion. As the person who had lived in Indiana (a pretty religious state though not part of the Bible Belt) the issue at hand is actually disrespect rather than religious affiliation.

That actually was the whole point of the Top Gear episode you have linked.

One of the few occasions you might have to pretend to be Christian would be a KKK Rally.

In most rural areas of the Bible belt if you are polite and respectful no one would care what is your religious affiliation or if you have any.

There are areas where you might be treated with extreme suspicion but this doesn't have to do with religion, but rather blood relations.

General recommendation when traveling anywhere is to avoid discussing any controversial subjects with locals. This include religion, politics, minorities rights, etc. Always divert conversation away from these subjects. If you're a tourist you're there to see, not to make a statement.

Also discussing other subjects, don't try to be smartass about local customs. You might for example despise NASCAR or handegg (aka American football), but you won't win any friends by trying to convince locals that F1 is true racing and soccer is true football.

This doesn't, however, mean that you need to pretend to be a follower.

In general, no, you do not need to pretend. Just be respectful.

There's no check at the border, nobody questions you. There's no official state religion, and the whole "Freedom of religion" thing is huge there.

Of course, wherever you go - in any country, there's going to be people who discriminate.

For the most part, however - most people, in every country, are more curious than judgmental.

There are situations where for social acceptance pretending to be of the right faith matters but that's all. It's not a safety issue other than in extreme cases (such as the KKK rally Karlson mentioned. I would NOT recommend shouting "Allah akbar" in a KKK rally--but I would suggest staying away in the first place anyway.)

Having traveled extensively throughout the U.S., I can say that overall it is a very safe place to visit, regardless of your religion. There are isolated incidents of issues, but statistically it's pretty small.

Freedom of Religion and Respect

"Freedom of religion" is an important cornerstone of the U.S., and most of the U.S. is very tolerant of other religions (assuming you are respectful of their beliefs).

Most people would be curious, rather than hateful.

Don't Offend the Natives

If you say offensive things about Christianity or Americans, that is a different story. When visiting any country, it is best to not offend the natives.

Most Non-European Foreigners are Assumed to be Non-Christians

Unless you are European, pretending probably would draw more attention to yourself, rather than less. Americans tend to assume that non-Americans and non-Europeans are not Christians, and might be surprised if you claim you are one.

A friend of mine from China is Christian, and some of the Americans I introduced her to were actually surprised she was Christian. They assumed she was Buddhist or something else. These same people also didn't realize that most Filipinos are Catholic.

So there really is no need to pretend to be Christian. Unless you're European or American, it is assumed you aren't one.

Avoiding Being Converted / Proselytization

The biggest issue some people have with openly not being Christian, is that some Christians will try to convert you. So pretending to be Christian will sometimes get them to back off. This is why many Americans pretend to be Christian, when they never really even go to church and are much more secular in nature.

Secular Society

The U.S. has a very secular society, that does not hinge on your religious views. Specific communities might be more religious than others, but overall, the economy and government policies are all secular. This usually creates a divide where people are secular in their daily business and in the workplace, but religious in their free time and at church.

Instead of pretending to be Christian, it might be better to just stay off the topic, and keep things secular.

Isolated Communities

There are isolated communities that don't take kindly to strangers (of any kind). Usually those are small backwoods towns that most tourists would never find themselves in. The issue in those towns is that you are a stranger, and being Christian wouldn't necessarily help you in that situation. They'd find something else to be annoyed at.

Overall

In most cases, there is no need to pretend. You may find yourself in situations where its best to keep your mouth shut and blend in, but that is true in any country, whether born there or not.

As long as you respect others, and stay out of bad neighborhoods, you will probably be fine, regardless of what religion you are.