Played through the main storyline, and then started a new game to do some more side quests and to give hardcoe Mode a try, and now it's not letting my character level up. It takes 200 XP to get to level 2; I'm currently at around 600 XP and still no little drum roll and level-up screen. Is that what makes hardcoe Mode hardcoe? I have to face deathclaws as a level one character with a 34 in Guns?

I'm thinking you already know this, but you can't level up around enemies, all hostiles must be dead in the vicinity before you can level up. If your in an area with no enemies around you at all, you've hit a glitch. Try going back to another save or restarting from scratch.

Not around any enemies, so it must be a glitch. I'll go back a few saves and see what happens.

I read in thehttp://fallout.wikia.com/wiki/Fallout_Wiki that if you skip the the tutorial quest with sunny smiles on your second or higher playthrough your game can glitch and you can't level up.
Not sure there is a fix besides restarting.

I can't remember which page on the wiki I read it though.