This argument is not interesting, because "there is no way to know" is a correct but useless answer for basically any physical question.
The best thing we know how to do is to look at the evidence that nature gives us, and try to construct a description of nature from them. We will have different confidence levels for different descriptions (e.g. deterministic vs. non-deterministic), based on how well they match the evidence, but we can never know for sure. Therefore the interesting question is how confident are we that nature is deterministic / non-deterministic.
> But the other thing to point out is that this question presumes an old idea about the passage of time, which is that things happen in a sequence A, B, C, D, ... and that if you are at C then D "has not happened yet". But if you look at relativity, this appears to be a naïve viewpoint.
In relativity, the notion you described is replaced by the weaker notion of causality. It says that for each observer things will still happen in a well-defined sequence, even though different observers will not agree on questions such as "did these two events occur simultaneously?". The way these statements can be consistent, is that events for which there is no agreement on ordering can never influence each other (this is a mathematical result).
Due to causality, the question of whether nature is deterministic is well-defined even when relativity is taken into account.