I’m also trying stick-breaking IBP for my research.
Too sample new stick-size mu from its conditional probability (eq.4.51 of the thesis), I recommend to use naive rejection sampler because it does not require log-concavity.
In concrete, in eq.4.51, the first and second terms
mu ^ {a - 1} * (1 - mu) ^ N
are regarded as an unnormalized pdf of Beta(a, N + 1). Then, the third term
exp(a \sum (1 - mu) ^ i / i)
has an upper bound
exp(a * Hn),
where Hn is an N-th harmonic number, since 0 <= mu <= 1 is guaranteed.
In summary, you can sample new stick-size mu using rejection sampler with proposal distribution Beta(a, N + 1) and constant coefficient B(a, N + 1) * exp(a * Hn). B(x, y) is Beta function.