I'm not sure how basset works, but if you can output the file name (not the HTML, the actual file) without the hash, you can use H5BP's rewrite rule to use the hash in HTML and route to actual file. This gives you the benefit of cache-busting without the pitfalls you're experiencing of being behind a load balancer.
There's an nginx version also as your server (or at least your load balancer) appears to be nginx.
Lastly, this if off-topic,
- Have you made sure that your sessions are "sticky" (or made sure your sessions are persistent on a central storage?). Load balancing tends to mess that up unless care is taken
- Every web request has the potential to look like its coming from the same user (aka your load balancer) unless the user's useragent/IP address is passed through to your origin servers.
Just mentioning that in case you run into it as well. It's completely off-topic to your Basset issue. It's something I've run into a bunch.