I have just looked in my books about the Habsburgs and in the one by Reiffenscheid "Die Habsburger in Lebensbildern", we have the following dates:
Claudia gave birth to two daughters, which both died very early: Anna Maria Sophie (10.09.1674- 21.12.1674) and the second one Maria Josepha Klementine (11.10.1675-11.7.1676).
Claudia herself died on April 8th, 1676, so I think her death is not directly in connection with the birth of her last daughter, but some months later, but I am sure she was weak from the births and maybe (not only but also) because of that not very healthy. If she had any special illness I do not know, but the main problems seem to have been the close relationship of her parents and ansistors, like often in Hapsburg marriages, the results are not very healthy descendants in many cases, you see this also when you look at the life dates of her children, Leopold I, her husband was like her a Habsburg and from her parents she had Habsburg and Medici blood from two sides (very closely), Claudia herself looks nice, but tenderly and not very strong in my view.
If anyone has further infos on the circumstances of her death it would be interesting.