So I'm going through the game as a pure, stupid, tough fighter. A Barbarian. Naturally, I join the fighters guild.

Quest 1: Kill some rats. OK, done.

Quest 2: Kill some egg poachers. OK, done.

Quest 3: Kill four (!) agents. Three are grouped together. Tough, yes, especially for my low level character but my battle axe has the last say. Done.

Quest 4: Collect a code book. OK, this is where I'm stumped. I have three options: Break the law and kill her, steal it (not really, sneak skills too low), or taunt her (not really, personality skills too low). So as a barbarian, I really only have one option. An illegal option. One that comes with a 1000+ septim cost for a quest that pays 50 septims.

Just for giggles, then, the next quest asks me to collect money from the Suran [censored]house. Once again, only three options, two of which I can't feasibly do because I'm a fighter.

Is the only way to advance to break the law?

Apparently you have missed out on a major aspect of how the game works.
Disposition.
The more people like you, the more they are willing to do what you tell them.

Bribe anyone who is giving you hassle. They will usually bend to your will.
Bribes don't fail as often as Taunt/Admire, so it's not too difficult to do. (a few bribes of 100 gold is usually enough)

Get her disposition above 70 and she just gives you the book.
Get her disposition above 75 and she just gives you the money.
Or, you can just pay the debt yourself (speak with Percius Mercius)

You could also refuse to finish the quest, that's a fourth option, I suppose. I'm not sure how that affects your progress in the Fighters Guild, though (none of my characters have ever joined that guild, I think).

EDIT: I don't really see a barbarian bribing anyone. It's a shame the Intimidate action doesn't also take Strength into account.

That's a valid option. You won't get any more quests from Eydis until the end but you don't need her to progress; you can go to one of the other quest givers.