"Having" and "As" both imply that the purse was in the process of or had recently been stolen. "With my purse stolen" has no implications about when it happened.
Both of those sentences sound unnatural. "I felt helpless after my purse was stolen" or "Because my purse was stolen, I felt helpless" sounds more like the way this would be said. I agree with Geostigma that "having" and "as" imply that it was happening at the moment.
If you say "With my purse stolen, I couldn't but feel helpless," you are implying that the condition of having your purse stolen is complete and that it leads to a feeling of helplessness. It's almost like saying "Since/because it was gone, I couldn't but feel helpless." So the choice of "As" really isn't incorrect if the "As" also means "since", which it can sometimes.
I think the confusion comes from the use of "As" in the context of the original exercise. The original "As" seems to indicate that the situation was currently happening. Almost like, "As it was being stolen," or "At the time it was stolen."
Your interpretation, which is not incorrect, feels more like "Because it was stolen," but regarding the context of the exercise, the "Having" agrees more with the "As". Your example assumes a finished state, while the exercise was maybe talking about the time of the situations occurence.
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