In my experience 5 months is a little early to even start trying to get a breastfed baby to sleep without the boob. That works better around 9 or ten months, if at all. (The nighttime feeding being the last my kids gave up, at age 13 months for my middle child and at 15 months for my youngest.)
That being said, I do sympathize with your desire to help establish a healthy bedtime routine. even if your baby continues to breastfeed herself to sleep, getting her to do it in her own room, and then settle herself post-tuck in (and more importantly when she wakes up periodically at night like all babies do) is a great idea.
You say you start her routine in her room. That's great. All you have to do is work on staying there. I found it worked best for me to sit and feed the baby right next to the crib, then I'd hand the baby over to daddy, he'd burp and then tuck in. If I did the tuck in, the boys would wake and want to nurse again. Daddy sometimes had to deal with a little fussiness. Daddy's do well to flex the patience and persistence muscles at times like these. It's easy to give a fussy baby back to mommy, but it'll set you back. Resist on all but the toughest nights, when she's overtired or not feeling well.
I'm not going to get into elaborate sleep training detail-there are plenty of other good answers on parenting se to questions about that specifically, but, to get you headed down the right path and answer your question, option 2 is my recommendation, based on your stated goal.
(And, BTW, if the baby is used to sleeping with you guys, you can make her feel more "at home" in her crib if you toss her clean crib sheet in between your sheet and blanket for a few days before making her bed with it. That way, it smells like your bed. She'll appreciate that more than the smell of whatever detergent you use. Scent has a powerful effect on babies, and grown ups too. Use it to your advantage!)