Impact of Connection Close vs Keep-alive
Solution 1:
First and foremost, yell. Loudly. At your vendor. For having a product not supporting the over-a-decade-old HTTP/1.1 protocol.
The impact of not having persistent connections is a major increase in loading time of resources. With keep-alive, a single TCP connection can be used to request multiple resources; without, a new TCP session (with a new three-way handshake - and, if you use SSL, a new SSL negotiation) is required for each and every resource on the page.
In practical terms, the impact will depend on the number of resources on a page, the round-trip time between client and server, and the number of concurrent requests a client's browser is making at a time (modern browsers run ~6ish by default). Lots of resources per page and distant clients will mean a very noticeable increase in page load times.
Solution 2:
Keep alive will greatly enhance the performance on both the client and server side. If possible do not disable it. The load balancer should work fine with keep alive turned on.