I've never done it, but I've heard that it tears up the alfalfa pretty badly. Like ARC55 said, it'll leave a lot more waste. I guess you could do it in a pinch, I always did my haying with a mower conditioner that I got used. If you go to winter equipment auctions, you might be able to find a used mower conditioner for a good price. Haying equipment always sells cheaper in the winter. Disc mowers are the best to cut hay, but they're also a lot more expensive. My old mower conditioner always did what I needed and it wasn't that expensive.
We've had to do it before. Although it works, it won't bale up the same way and you'll have lots of messy bits and pieces. You can square bale it, but it's tough to do. We just stacked ours and covered it up. It seemed to take longer to complete, but maybe that's because we weren't used to doing it that way. We had more time than money, though, so it was worth it to us. As for the waste, we rotated our livestock out that way to clean it up and make use of what was dropped. That might not work for everyone. I should say that my husband and I were younger then. It's tough physical work. Given a choice, I would use the proper equipment for the job, but when needs must . . .
I think you'll be fine as long as you go into it with realistic expectations.