I don't think Chaitin is committed to either a (Turing) computational or hypercomputational universe.
Quoting Chaitin from a live panel discussion on digital physics:
Look, I’d like to be aggressive about this — the best way for
me to think about something is to make claims that are much too strong
(at least it brings out the idea). So the universe ...
This is my point of view on the topic, very different from yours : time is not a measurement.
As regards "TIME", it should be considered whether an object has the NON-evident capability to MUTATE. When this capability exists, then we can say that TIME EXISTS for that object. During mutation old states of the object are replaced by newer ones and lost forever....