what would you consider to be meaningful evidence that non-human animals suffer in a morally relevant way?

I freely admit that animals can and do feel pain. "Suffer" is a complicated word and it's possible to debate whether it can properly be applied only to humans or not only. However for simplicity's sake I'll stipulate that animals can suffer.

Now, a "morally relevant way" is a much more iffy proposition. It depends on your morality which is not a matter of facts or evidence. In some moral systems animal suffering would be "morally relevant", in others it would not be. No evidence would be capable of changing that.