Valid token, valid request, no response using page access token - facebook-graph-api

Question

I have made a web app that tries to create a page access token that never expires. It does this because it's all written in PHP, and needs access to the data without asking the user to login repeatedly.
The Facebook docs say this:
To get a longer-lived page access token, exchange the User access token for a long-lived one, as above, and then request the Page access token. The resulting page access token will not have any expiry time.
And so I have written the following PHP:
// get our current access token
$accessToken = $session->getAccessToken();
// make it a long-lived access token
$longLivedAccessToken = $accessToken->extend();
// find the list of pages we have access to
$request = new FacebookRequest($session, "GET", "/me/accounts?fields=name,access_token,perms");
$response = $request->execute()->getGraphObject()->asArray();
// get the first page access token
$pageAccessToken = $response["data"][0]->access_token;
This seems to work: the access token debugger says the token never expires and is valid right now. I can even use this token to do many things.
But my token cannot do some things, and I don't know why.
Where I am hitting the problem is reading the /insights/page_fans_country/lifetime end point.
If I use /mypage/insights/page_fans_country/lifetime I get a response. That is, mypage is the page that was used to generate the permanent access token.
If I use /otherpage/insights/page_fans_country/lifetime I get no data.
As far as I know, this is public data – if I generate a new access token in the Graph API explorer, I get valid data for both requests.
So:
1) Am I requesting the permanent page token correctly?
2) Is this just a limitation of (permanent) page tokens – that they cannot access some public data?
3) If so, is there a correct way of accessing this data using tokens that don't expire?