[docs] Document the client checks that bypass the agent network endpoint

A few client-side checks call their vendor directly instead of following
the configured base URL, so they fail on exactly the locked-down networks
Agent Network is built for while inference keeps working. Fast mode reports
a connectivity error, or reports itself disabled by the organization when
the agent holds only a proxy-issued token, and model discovery stays off
until it is turned on explicitly.

Name the variables that settle each case, and say plainly which ones
allowing direct egress does not fix.

agent-network/README.md

@@ -40,6 +40,35 @@ You can then use this private endpoint to configure your AI agents, whether that
 Full step-by-step setup:
 **https://docs.netbird.io/agent-network/quickstart**
 
+## Client settings that don't follow the endpoint
+
+Most of an agent's traffic follows the base URL you hand it, but a few
+client-side checks call their vendor directly and never reach the proxy. On a
+network that blocks direct egress they fail even though inference works, so
+they are worth setting once when you roll the endpoint out.
+
+For Claude Code:
+
+- **Fast mode** checks availability against `api.anthropic.com` rather than the
+  configured base URL. Set `CLAUDE_CODE_SKIP_FAST_MODE_ORG_CHECK=1` when the
+  agent authenticates with `ANTHROPIC_AUTH_TOKEN` alone (the usual shape when
+  the proxy injects the real provider key) or when a TLS-inspecting proxy
+  answers the check itself. Set
+  `CLAUDE_CODE_SKIP_FAST_MODE_NETWORK_ERRORS=1` when the network refuses the
+  connection outright. Fast mode is an Anthropic-API feature, so it is
+  unavailable on a Bedrock- or Vertex-backed endpoint whatever you set.
+- **Model discovery** is off by default. Set
+  `CLAUDE_CODE_ENABLE_GATEWAY_MODEL_DISCOVERY=1` for the picker to list the
+  models your policies authorise; the proxy filters the response to that set.
+  The client gives discovery a three-second budget and treats any redirect as
+  a failure, so the endpoint must serve `/v1/models` directly.
+- **The WebFetch domain safety check** also calls `api.anthropic.com` directly
+  and is unaffected by the variables above.
+
+Allowing direct egress to `api.anthropic.com` covers the network cases but not
+the credential one, where the check reaches Anthropic and is rejected because
+the agent presents a proxy-issued key.
+
 ## Architecture 
 
 Agent Network is built on two existing NetBird capabilities: