..made it, no problems.
This thread on the LOTRO forums was helpful, although I didn't follow all the advice there. In particular, I did a lot of lower level quests, simply because I was questing through The Shire for the first time (my other two characters are a dwarf and a man), and wanted to enjoy all the content. Doing quests below your level is obviously quite a safe way to level, but also slow, and for many people, the Undying attempt goes wrong not due to trying to do something difficult, but simply bad luck like a network dropout or game crash while in combat. So the slower you level, the more risk there is of that happening.
But yeah, I just ground it out, solo'd all the group quests in The Shire (although outlevelling them when I did), went to Bree-land at about level 17, and popped out quests there until I hit 20. Oh, and I had top-quality crafted weapons all the way due to my Champion being a weaponsmith, upgrading to a new crit crafted weapon every two levels. So my damage output was pretty high. That went a long way towards reducing the risk of multi-enemy fights.