Welcome Raul.
Kiwi's meant a septon, but he's absolutely right. Tyrion married secretly... so it can be done. And I think a septon would consider any request by the Crown Prince to be a command.
But the implications to the throne and inheritance of taking multiple wives and siring sons is immense. I would think that if Rhaegar and Lyanna had made their marriage public, then the Starks and the Targaryens may have been able to settle on a diplomatic arrangement rather than war. Remember, Lyanna's abduction was the cause of Brandon's public denouncement of Rhaegar which led to Brandon's arrest, etc. The Martells may not have liked a second queen, but the Martells would have been a far easier problem to deal with than the Arryn-Baratheon-Stark-Tully alliance was.
In another thread, TK accused me of double talk. I will do this now.
On the other hand, many people with vital information were killed during Robert's Rebellion. For instance, the pyromancers responsible for the Mad King's orders to store wildfire in order to blow up King's Landing seemed to have all been killed, most notably Lord Rossart. So if Rhaegar and Lyanna wed just before he rode to the Trident, who would have known this? The septon (this could have been Pycelle), the Kingsguard guarding Lyanna, and Ashara Dayne could have all known. Ashara and the Kingsguard are all dead. The septon could have died or been too frightened to come forward with the tale after the Targaryens' utter defeat. If it was Pycelle, he'd probably have leaked it to Tywin Lannister.
I've always assumed that R+L=J was on the wrong side of the sheets. Lyanna's "promise me, Ned" was explained away by Robert's irrational loathing of all Targaryens. But could Robert have somehow been persuaded to see Jon as the last and best vestige of Lyanna? Could Robert have been coaxed to see only Lyanna in Jon and not see Rhaegar? If Jon was born of rape, then perhaps Ned might have been able show Jon in a positive light to Robert. But if Lyanna married Rhaegar, then she not only willingly went from Robert, her betrothed, but she married without even breaking it off with him. To add insult to injury, Jon could have been considered Aerys II's heir and not Viserys... Jon was either the rightful king or Viserys' heir. Jon would have had to die no matter what Robert felt for Lyanna.
"Promise me, Ned" takes on an even more desperate tone now.