You're right, curl won't browse your website like a real user with a real browser so you'll need something else (if you care about that differentiation).
Remember that JS executes in the browser so this won't necessarily add load to your server (this of course depends on if your JS invokes a call to the server to do more work, like an ajax call). Do you care about what happens once the page hits the user, or are you testing the performance of your server? Either way, you have a few options for performance/load testing.
You could use something like phantomjs (useful if you need that fully rendered webpage, or you need to interact with the JS...this gives you what curl can't), but even then you won't be able to easily simulate lots of users from different locations.
You could look into beezwithmachineguns but that would probably require modification to properly load the JS like a real browser would.
Since you don't have the time, try an online service that already does what you want. I've used https://www.blitz.io/ in the past and since it's software as a service there's nothing to download or configure so it's easy and painless to get going.
If you have enough hardware you can consider using Selenium Grid.
Or, consider the following commercial tools on a limited license basis
If you have reached the point where the financial risk is sufficiently high to warrant performance testing then what you will find is that if you measure the financial impact of failure against even the most expensive commercial tools available that the cost of failure dwarfs the cost of any of the available market tools.
You can also still achieve a negative ROI with a free or open sourced tool if you give it to a person without the skills to achieve success, similar to handing the keys to an exotic italian sportscar to someone who doesn't believe in cars and who has ridden busses and trains their entire adult lives. Car very capable: Driver, not so much.