Driving with a broken sway bar link is risky, especially if you do a lot of cornering or driving in traffic. The sway bar (also called anti-roll bar) helps keep your car stable during turns by distributing weight evenly across both wheels. When a link is broken, that stabilization fails on one side, which means your car will lean more heavily into turns and your suspension becomes less predictable. The clunking noise you're hearing is the broken link moving around, and that's your signal something's seriously compromised.
The main danger is that you'll have reduced control in corners - your car might roll more than it should, or the suspension geometry gets thrown off in ways that affect braking and handling. If you're in an emergency situation (sudden swerve, sharp turn at speed, emergency braking while turning), that's when a compromised sway bar really bites you. Driving cautiously on straight roads for a month might seem okay, but you never know when you'll need that stability.
Before committing to waiting, have you gotten a second opinion on the price? $300 seems reasonable but varies by vehicle and whether it's just the link or if there's other damage. Also, is this the only thing that needs fixing right now, or are there other suspension issues? If you absolutely must delay, at least drive extra defensively - slow down on curves, avoid heavy traffic, and don't brake hard while turning. But honestly, if you can swing it sooner, you should.