Here's the solution I came up with:

private const int LDAP_OPT_TIMELIMIT = 0x04;

[DllImport("Wldap32.dll", CallingConvention = CallingConvention.Cdecl, EntryPoint = "ldap_set_optionW", CharSet = CharSet.Unicode)]
private static extern int ldap_set_option([In] IntPtr handle, [In] int option, [In] ref int inValue);

private static void SetLdapConnectionBindTimeout(LdapConnection conn, int timeoutSeconds)
{
    // We need the underlying LdapConnection handle; that's internal, so reflection here we go.
    var handleField = typeof(LdapConnection).GetField("ldapHandle", BindingFlags.NonPublic | BindingFlags.Instance);
    var handleWrapper = handleField.GetValue(conn);

    // That handle object is itself a wrapper class around the IntPtr we actually need.
    // The wrapper class is internal, and so is the IntPtr, so more reflection.
    var internalHandleField = handleWrapper.GetType().GetField("handle", BindingFlags.NonPublic | BindingFlags.Instance);
    var internalHandle = (IntPtr)internalHandleField.GetValue(handleWrapper);

    // Now we can set. 
    ldap_set_option(internalHandle, LDAP_OPT_TIMELIMIT, ref timeoutSeconds);
}

This works, but I certainly don't love all that reflection. Or the DllImport, although the .NET library uses that under the covers anyway, so I feel like that's not as big a deal.

Per the comments below, it's also good to note here that due to the dependency on Wldap32.dll, this would appear to be Windows-only and not suitable for cross-platform.