Forward DNS request to my work's jump server

Partial answer:

You can't selectively forward DNS requests for specific hosts in general (like Foxyproxy can handle specific requests for specific hosts or paths).

The only thing you can do is to run a DNS proxy somewhere that would answer requests for specific hosts itself, or forward requests following specific patterns to specific servers.

One popular DNS proxy is dnsmasq. You can configure it to forward DNS requests to different servers based on a pattern.

So you'd have to run dnsmasq on your local machine, use it for DNS resolution on your local machine, and configure it to use both your ISP's DNS server and the jump host DNS server as upstream DNS servers. That will require a bit of fiddling to get the configuration right, in particular to make it pickup both a dynamically given DNS your server (from your ISP) and a statically given one.