Is it possible to determine how k was generated in this situation (e.g., weak PRNG, timestamp-based, or something else)? And could we potentially reproduce the same k with the right assumptions?
Unless r has an obvious value (eg. 1*G) it is going to be very difficult to guess how it was chosen.
The only method is to use a search engine to see where the address/transaction is mentioned and see the reason there.
In the past when any of these broken keys were found, the owner is asked what wallet they were using and by analyzing the behavior and/or looking at its source code the broken part was found. For example when the users said they were using blockchain.info the source code was analyzed and it was found that they were using random.org to generate the ephemeral key and the site was returning an error message which the app still used as input which ended up being the same for everyone!