Archbishop Ballestrero was certainly 'church,' but I think certainly not 'science.' His involvement with the science of the shroud was coincidental with his being the Archbishop of Turin at the time. After the dating, he was apparently happy to accept the 14th century date, although in old age he was quoted as claiming the whole process to have been a masonic plot.
Giovanni Riggi was 'science.' A stamp collector by trade, his interest in the shroud was technical rather than religious. I don't know what his view was of the date of the shroud, before or after its sampling. Gove suggests he was writing a book deploring the use of only one sampling site. That may have been what he was arguing with Gonella about, although his original involvement with the shroud came about because he was a personal friend of Gonella's.
Luigi Gonella was definitely both 'church' and 'science,' being both a Professor of Physics at Turin Polytechnic and Scientific Advisor to the Archbishop. However his interest in the shroud during 1988 appears to have been mainly political, and mostly typified by his antipathy to Carlos Chagas, of the Pontifical Academy of Science, who 'outranked' him until Gonella managed to have him ostracised. Gonella appears to have accepted the C14 date, although had hoped for authenticity.
Of these three, only Gonella was involved in meeting the other scientists. It could be said that he 'represented the church.'