Really breaking out the hard questions these days, eh?

The short answer is that to my knowledge there is no significant documentation of a Papal reaction to the Taiping uprising, or if there is it has simply not been published or featured in any published material of which I am aware.

That is not to say there was not some sort of Catholic response, although it was a very mixed and individual one. The 1853 report by Bishop Francisco Xavier Maresca, the apostolic administrator in Nanjng, was notably catastrophising in tone, reporting (although it seems he had made his escape long before) the confiscations of property and outright massacres of Catholic converts in Nanjing and Yangzhou when the Taiping burst on the scene that spring. But later in the year, French consular official Mgr de Bourboulon was sent to Nanjing to attempt to establish diplomatic contact regarding these incidents, and although he received a rather frosty reception he was ultimately assured of the converts' safety and convinced of Taiping success. His contemporary, the Apostolic Vicar of Jiangxi Dr Louis-Gabriel Delaplace, was also quite positive that a Taiping victory would lead to Catholic toleration but feared that a Qing victory would lead to a new round of repressions, like with the Rites Controversy of the early 18th century. Delaplace was moved to Zhejiang in 1854, and his successor, Francois-Xavier-Timothée Danicourt, had similarly high hopes when writing in 1857, although he maintained quite a lot of reservations about the Taiping's methods. I covered the Catholic missionaries' opinions in a liittle more detail here. On the whole, however, ovre time Catholic opinions of the Taiping were mostly negative, due in no small part to the Taiping generally failing to recognise Catholics as fellow Christians, and so there were numerous reports of Taiping troops killing Catholic converts even after 1853.

There did not end up being a situation like the Ex illa die kerfuffle under the Kangxi Emperor, probably because there was both limited missionary contact and because the Taiping were commonly thought to have Lutheran rather than Catholic origins, although I would be surprised if there were not some sort of papal response that simply has not been drawn upon by historians of the period. I apologise for not being of more help, but perhaps there is some way of accessing Vatican records for this time period.

If you're interested in the broader Catholic response, I'm not sure there's much published work on the subject (I have, sadly, not yet been able to get my hands on a reasonably-priced copy of Carl S. Kilcourse's Taiping Theology), but Prescott Clarke and J.S. Gregory's source collection Western Reports on the Taiping has a reasonable number of translations of reports and letters by the Catholic missionary community.