Any reason to doubt physicalism? Is there anything in science that says “Huh well that seems to not have any basis in the physical at all and yet it exists”

If it had no basis in the physical, then what would it mean to say that it "exists?" How you define "existence" is a very big philosophical question. Excuse me while I nerd the fuck out about something.

Physics tells us that the observable universe is 93 billion light years in diameter. However, we can sometimes observe objects leaving the observable universe. This is because of complicated physics reasons:

Physical space is expanding with time. Everything is getting farther apart from everything else, and the more distance there is between two points, the faster the space in between them is expanding. At a sufficiently large distance, the rate at which the distance between the two points is increasing, is faster than the speed of light. Neither point is actually moving faster than the speed of light, it's only the space between them that is expanding. This might be hard to understand, but think of it as if you drew two dots on a balloon and then inflated it.

Once an object gets far enough away from us that the space between is expanding faster than the speed of light, it becomes impossible for us to make any further observations about that thing. This is actually what defines the bounds of "the observable universe."

So, what happens to objects that leave the observable universe? Strictly speaking, it's impossible to say. Intuitively, we would expect that they're still there doing their thing and obeying the same physical laws as when we could observe them. But, if you told me that the stars simply vanish, or that they magically transform into butterflies as soon as they leave, there's no evidence that anyone could ever produce that would falsify that belief, because, by definition, there is no way to observe what happens outside of the observable universe. If we are defining what exists based on what is physically observable, then it follows that things outside the observable universe do not exist, even if it really seems like they should.

My conclusion from this line of thought is that existence is a relational property. I am not prepared to reject the idea that a thing has to be in some way observable in order to exist, but in that case, nothing can exist in isolation. Because for a thing to be observable means that there must exist a being which could observe it. This could be said to contradict physicalism, because physicalism would say that the material world exists regardless of our senses. I would say that the physical world only exists so long as there are beings capable of sensing it, and, should all sentient beings ever become extinct, the physical world would no longer exist in any meaningful sense.