I only have gene expression values at the species level…

That means you can’t distinguish expression levels from other characteristics of the species. Keep that in mind.

We therefore want to measure whether the expression of that gene in a species is correlated with the observed effect of the compound, since that will tell us the degree to which the targeting of the compound by the antibody matters…

As Glen_b said in a comment, "I don't think correlation generally tells you how much something matters (at least not directly).”

Is a Pearson correlation appropriate for data like this with many repeated values?

Having multiple observations at the same expression level isn’t substantially different from a designed experiment in which you, say, test several distinct doses of a drug and have dose as the “predictor” variable. There’s no general problem with multiple ties in a “predictor” variable, the way you are using expression level.

You certainly can calculate a Pearson correlation here. Problems might arise with the usual ways of testing the “significance” of the correlation (in terms of the difference from no correlation), which work best with a bivariate normal distribution. There are other ways to perform such tests, if you wish (bootstrapping or permutation tests).

It’s probably not appropriate, or at least it’s less than optimal, however. Correlation only picks up linear associations: between values with Pearson correlation, between ranks of values with Spearman or Kendall (which are often better choices). That can leave a lot of useful information unexplored.

If you have enough different expression levels (species), it would be more informative to model the continuous phenotype flexibly as a function of expression, for example with a regression spline. You can then evaluate how much of the association is linear (in the expression/phenotype scales of the data) and whether non-linear associations are substantial. There's a further advantage of a regression approach: you could include other characteristics or data on the species or on individuals that might also be associated with phenotype (age, weight, sex, expression of other genes…).

Quantifying the association between phenotype and expression that way provides a start at evaluating how much this association “matters,” however you are defining that word.