In practice, you’re far more likely to agree with fellow workers than with bosses. You have material interests largely in common, not largely opposed. That’s why unionised workplaces and industries have measurably better pay and conditions.
There are other bosses. They are different. And my industry is superior to most unionized industries so I don't care.
It is true that the workers of rich countries are “bribed” with less exploitation to pit them against those in or from poor countries. But even their condition can be improved, so it’s possible to build solidarity. I’ve seen it myself.
Ultimately, all value is created by workers and that is where profits are extracted from. If we can foster solidarity with one another, we can force bosses to keep less of our labour as profits.
It’s individualism that is foolish, since it lets the bosses divide us. Collective action is much more effective.
But there is a simple way to solve this. If your union has, as a bylaw, "It is an immutable rule of this union that it shall never act to decrease or keep constant the number of immigrants to the USA. If this union acts in that manner, it shall be immediately dissolved and all membership rolls shall be destroyed after refunds for that month's membership fees", then I won't oppose it.
Considering that we are to act in solidarity, all I'm asking is for that measure of good faith. Just put down the knife and go stand in front. That's all I'm saying.
I'd recommend you contact some local trade unionists, you'll find that they're overwhelmingly pro immigration and anti-racist. You'll also find that marginalised groups (including immigrants) are vastly over-represented in the labour movement in general, since we are exploited more.
The fact that they won't, and instead use toothless codes that can be repealed when things change make it clear they don't believe this is an immutable part of them but is a marketing ploy.
How would such a "poison pill" even be enforced? If the majority decides something they can always do it, worst case by creating a new organisation which they then all join.
The US is very racist and such attitudes persist even in the labour movement. Historically, US reformist unions used to exclude non-citizens from membership, which is why the IWW started organising all workers including immigrants. The result of this democratic struggle is that US unions now organise all workers.
You're not losing anything by joining a union. It's possible that you may not gain as much as you'd like because of bigotry where you work, but that can also be fought against.