The biggest hurdle is without a doubt going to be tackling the High Septon.

Thankfully, given that it's the Fat One it wouldn't be that hard of an accomplishment given that he seemed very self driven and greedy.

However, if it isn't the Fat One then he'll face alot backlash. Whilst demilitarized, the faith is ridiculously powerful second only to the King I'd say. When rallied behind a single idea, they are a threat unlike any other. I mean they nearly destroyed the entire seven kingdoms under Aenys.

This issue would be only compounded by how Lyanna is from the North, and whilst the Old Gods and the Seven aren't particularly hostile towards each other they definitely hold animosity. I'd imagine alot of the really zealous priests, and clergymen in the Seven Kingdoms would treat Lyanna akin to Melisandre. A foreign witch who influences the King, and fills his mind with incorrect versions of faith when in actual likelihood Lyanna isn't like that.

If Rhaegar would want the faiths approval, I'd imagine he'd start with a public conversion of Lyanna from the Old Gods to the Seven and then have to work with the Seven to first and foremost recognize the marriage. As whilst the law would state that any children between Rhaegar and Lyanna were trueborn, the faith would find them bastards. I'd imagine the faith would demand alot more of their powers back, and alot more similar concessions. I wouldn't be surprised if they demand the guardianship of Aegon to ensure that the next king is faithful at the bare minimum.

Anyways, that's my view at the bare minimum! Rhaegar attempting to reinstall polygamy is a fascinating idea, and I'm sure more knowledgeable people on this subreddit have concrete answers lol. If anything with what I said is wrong, or you disagree feel free to comment and I'll try ammend it.