Obviously I don't intend to take a stance on the math-qua-math question of P vs. BPP. But to the extent that someone has to assert that an algorithm's good BPP-related properties only work for an RNG rather than a PRNG, and there's no intelligent adversary of any kind involved in the system, I have to question whether this could reasonably happen in real life.

If your question is "Is there a known practical case not involving an intelligent adversarial environment where the use of a cryptographic PRNG or even a true RNG rather than a non-cryptographic PRNG is warranted?" Then the answer is no.
In fact, this is the reason why it is conjectured that P = BPP.

However, given the rest of your comment, it seems that you are referring to how we understand the theoretical properties of algorithms:

What I'm trying to point at generally is that unless I have an intelligent adversary I don't want my understanding of a piece of code to depend on whether a particular zero bit is "deterministic" or "random". I want my understanding to say that the code has just the same effect once the zero is generated, regardless of what factors generated the zero; I want to be able to screen off the "randomness" once I've looked at the output of that randomness, and just ask about the effectiveness of using a zero here or a one there. Furthermore I distrust any paradigm which doesn't look like that, and reject it as something I could really-truly believe, until the business about "randomness" has been screened off and eliminated from the analysis.

If we are talking about understanding the theoretical properties of many useful randomized algorithms, I'd say that we can't "screen off" randomness.
Even if the algorithm is implemented using a PRNG with a constant seed, and is thus fully deterministic at the level of what actually runs on the machine, when we reason about its theoretical properties, whether it is a formal analysis or a pre-formal intuitive analysis, we need to abstract away the PRNG and assume that the algorithm has access to true randomness.

As it was already pointed out, if we were perfectly rational Bayesian agents, then, we would always be able to include the PRNG into our analysis:
For instance, given a machine learning problem, a Bayesian agent would model it as a subjective probability distribution, and it may conclude that for that particular distribution the optimal algorithm is an implementation of Random Forest algorithm with the Mersenne Twister PRNG initialized with seed 42.
For a slightly different subjective distribution, the optimal algorithm may be an implementation of a Neural Network trained with Error Backpropagation with weights initialized by a Linear Congruentlial PRNG with seed 12345.
For another slightly different subjective distribution, the optimal algorithm may be some entirely different deterministic algorithm.

In practice, we can't reason this way. Therefore we assume true randomness in order to say meaningful things about many practically useful algorithms.