I'm hoping to find more information about the "Chair of St Peter" or sometimes known as the "Throne of St Peter", I've found art which has similar types of figures carved on the chair of the Pope, which I suspect may be supposed to imply this special chair for the pope. Apparently it was replaced sometime around the mid 1600s and there is an incredibly ornate one now. I suspect the VIeville and Tarot de Paris are both suggesting this type of chair with their use of the figures in the throne. I'm away from home for the weekend, but will find a couple images early next week.
I have never heard of the chair of St.Peter being decorated with sphinx's before!? (presuming you agree it is the figure of a sphinx, the image seems pretty clear to me). Not sure it is even a part of the chair (the figures on the papesse do appear to be part of the chair I would agree). Are you suggesting the papesse chair figures a frontal view of the popes sphinx seen from the side?
The sphinx took on new meaning in the 17th century through the work of Kircher, whose work on egyptology made him a sort of international superstar of his period, and much of which was achieved through papal patronage. Thorugh him the sphinx also became a symbol of the hieroglyphic 'riddle'.
Bernini's altar also inspired the style for a later work in the basilica, the Chair of St. Peter, an imposing piece that decorates the apse of the basilica. Above the marble base, there are four figures of the Greek and Latin Church: Anastasius, John Chrysostom, Ambrose and Augustine.*
Above those saints are two angels and a throne symbolizing the authority of Peter. Three bas-reliefs tell the story of three key encounters Peter had with Our Lord: the handing over of the keys of authority in the Church, the washing of the feet, and Jesus, after the Resurrection, asking Peter three times, "Do you love me?"
*The bronze columns of the baldachino that is said to here to have inspired the style for the chair rest on marble pedestals, and each of these is decorated with the coat of arms of Pope Urban VIII. Bernini, it is said, having heard that one of the Pope's nieces was pregnant, sculpted the face of a woman in various stages of pregnancy and childbirth on the sides of the four pedestals.
Pope Urban VIII was also one of Kircher's patrons;)
The wooden chair Bernini's reliquary houses was viewed in 1867 according to the catholic encyclopedia:
"In 1867, however, on the occasion of the eighteenth centenary of the martyrdom of the two great Apostles, it was exposed for the veneration of the faithful. At that time the Alessandri brothers photographed the chair, and that photograph is reproduced here. The seat is about one foot ten inches above the ground, and two feet eleven and seven-eighths inches wide; the sides are two feet one and one-half inches deep; the height of the back up to the tympanum is three feet five and one-third inches; the entire height of the chair is four feet seven and one-eighth inches. According to the examination then made by Padre Garucci and Giovanni Battista de Rossi, the oldest portion (see illustration) is a perfectly plain oaken arm-chair with four legs connected by cross-bars. The wood is much worm-eaten, and pieces have been cut from various spots at different times, evidently for relics. To the right and left of the seat four strong iron rings, intended for carrying-poles, are set into the legs. At a later date, perhaps in the ninth century, this famous chair was strengthened by the addition of pieces of acacia wood. The latter wood has inlaid in it a rich ornamentation of ivory. For the adornment of the front of the seat eighteen small panels of ivory have been used, on which the labours of Hercules, also fabulous animals, have been engraved; in like manner it was common at this period to ornament the covers of books and reliquaries with ivory panels or carved stones representing mythological scenes. The back is divided by small columns and arches into four fields and finishes at the top in a tympanum which has for ornamentation a large round opening between two smaller ones. The tympanum is surrounded on all sides by strips of ivory engraved in arabesques. At the centre of the horizontal strip a picture of an emperor (not seen in the illustration) is carved in the ivory; it is held to be a portrait of Charles the Bald. The arabesque of acanthus leaves filled with fantastic representations of animals, and the rough execution of the work, would make the period of this emperor (884) a probable date. What still remains of the old cathedra scarcely permits an opinion as to the original form. In any case it was a heavy chair made of plain, straight pieces of wood, so that it cannot be considered a sella curulis of Pudens, as earlier tradition held it to be. If the four rings on the two sides belong to the original chair (Ennodius of Pavia about the sixth century used the term sedes gestatoria as an expression universally understood in reference to this chair), then it was probably an ordinary carrying-chair, such as was commonly used in ancient Rome.
end quote from http://www.newadvent.org/cathen/03551e.htm