I cannot imagine that the pope had all that much influence on Austrian affairs in the early 20th century and even if he had agreed, it could have led to problems in, for example, the Hungarian part of the Empire, where such a change might have been viewed as a heaven-sent excuse to break away and become independent (the same could apply to any of the other parts of the Empire). The question of duress is also tricky, as in point of fact Emperor Franz-Joseph made a HUGE concession to Archduke Franz-Ferdinand, as other members of the family who married "below their station" lost everything. Take the case of Franz-Ferdinand's youngest brother, Archduke Ferdinand-Karl. When he married Bertha Czuber in 1911 he was excluded from the dynasty, lost all his titles, ranks, privileges, honours, orders and decorations, and the couple were officially banished from Vienna. Archduke Franz-Ferdinand was one of his fiercest opponents (which seems most ironic in view of his own 'unequal' marriage). Only the elderly Emperor mitigated the harsh punishment by authorizing a small annual allowance from the Habsburg Family Fund. Ferdinand Karl then became known as plain "Mr. Berg", a pseudonym sometimes used by his father Karl-Ludwig. After the assassination at Sarajevo in June 1914 Ferdinand-Karl Berg was granted a special dispensation to travel to Austria to attend his brother's funeral at Schloß Artstetten. He died in Munich on 12th March, 1915 aged 47 years and was buried at the cemetery at Untermais near Merano. Archduke Johann/Johann Orth is another example.
Had Franz-Ferdinand survived to ascend the throne I have no doubt that Sophie would have been treated with all due respect (as she was treated by Kaiser Wilhelm II) but as to whether Franz-Ferdinand would have attempted to change the status quo is a moot point. What seems more likely (at least in my opinion) is that he would have tried to do something to ensure that his sons obtained an appropriate title/position...