From your comments on another answer:
Since the idea is that this is a trial, thus the reduced budget, I'm wondering if there's a professional, diplomatic way of agreeing to the redesign, but not granting them commercial use and only providing reduced res jpgs for review.
I wouldn't play that game. I take a more cynical approach. It sounds like they're too cheap to hire a professional to really flesh out a proper logo in the first place. As others have mentioned, startups are notorious for testing the waters. You have reservations about giving them decent artwork without proper attribution. This isn't going end well.
If they are serious, I would counter with a more permanent position, say a month. As someone who's sat on both sides of the hiring table, it's hard to judge anyone's performance on one single project, especially in identity work. What's next, companies looking for typographers to flesh out a custom typeface?
If I was the CEO, he's probably looking at just ideation. They need to be straight with you at what they want and how they're going to use the artwork. Are they looking for ideation or a complete, fleshed out logo? We all know there's a huge gap between the two and if I were going to bet, they're looking at a few options. Whether there's an actual designer position available remains to be seen.
Now that they have your ideas, and thus artwork, would you be OK with them using that idea(s) and running with them, assuming your hourly rate was paid? Even if you stand behind a contract, that doesn't mean they're not going to breach it and have some less skilled designer take as swing with your ideas. What are you going to do, take them to court based on a copyright clause? That's not something you want to deal with.
You're also assuming they're actually going to use your logo. Who says there's not other "designers" they've sent a similar offer to? They're going to want full rights to whatever you provide to them, no question. If you can get more out of them, do it, but I agree with go-meek's answer: don't base this on the grounds of future work. If you're looking for that, then they should also be open to a more permanent position: say for a month while you complete the project properly.
Let's say you do get hired on based on your concepts, would you be OK with taking someone else's logo—the other designers you were "competing" with—and playing with it and possibly "combining" those elements with whatever you had? That's another possibility as well.
for $300, I couldn’t provide the same level of research, multiple rounds of revisions, etc. I’m also a little uncomfortable with the idea of a $300 trial project resulting in a mark that represents the entire company.
Go read, or re-read Chapter 6 in Mike Monteiro's Design is a Job. This is a classic case of a client trying to break your process. Don't fall for it.