This doesn't seem related to Pi-hole.
It's not entirely clear whether you are simply trying to browse using TOR from your LAN, or whether you are trying to access a public website you are hosting on a machine in your LAN network from another machine in your LAN network
Regardless, the following holds true if you are using a public hostname to browse a website with TOR:
When you are surfing with TOR, DNS requests for public remote domains are not made by the client browser at all, but rather delegated to be performed by TOR's exit node (see also Tor Protocol Specification, 6.4. Remote hostname lookup).
If your connection seems slow, you probably are routed through a slow or overloaded node, or maybe a high latency one that is geographically far away from your actual location. And of course, the website you are visiting could also be overloaded, or even try to reject connections from an exit node.
You could try to improve access to or performance of a website by requesting TOR to establish a new circuit for that site (effectively asking TOR to use a different exit node).