CloudFlare is pretty great, but as you noted, you can’t use WebSockets unless you’re on their Enterprise ($$$) plan. You may want to look into Amazon’s CloudFront, though CloudFlare is more feature-rich.
If you went the CloudFlare route, worst case scenario you could just disable WebSockets by using the environment variable DISABLE_WEBSOCKETS and setting it to 1. If you use a CDN like CloudFlare or CloudFront, make sure you also set HTTP_FORWARDED_COUNT to 1.