Sizing a Cuckoo filter for a subset of elements
As discussed in the comments it turns out that, yes, I need to size my Cuckoo filter for the set of every existing name. That being said, it is not much of a problem in the end if I only send the buckets with fingerprints in it, which shouldn't be more than 600.

In addition to that, my attempt at a self answer was very wrong. The probability that there is another name colliding with the name Anna is actually:

$$P\left(\bigcup_{i = 1}^n A_i\right) =  \sum_{k = 1}^n (-1)^{k+1} \sum_{1 \le i \lt j ... \lt k \le n} P(A_i \cap A_j ... \cap A_k)$$

This is the inclusion-exclusion principle. Because of my hypotheses, the probability can be simplified as:

$$P\left(\bigcup_{i = 1}^n A_i\right) =  1 - (1 - p)^n$$

As highlighted in the original paper, we also have to take into account that $$2b/2^f \le \epsilon$$.