How to query from a linked server via a linked server in sql server?
Unfortunately, with the conditions you've placed on your question, there is no way to accomplish what you want.
SQL Server would need to support 5 part naming such as:
SELECT *
FROM server1.server2.database.schema.table;

Which is clearly not going to work.
If you're not afraid of a little dynamic SQL, you could use something like this:
DECLARE @cmd nvarchar(max) = N'
DECLARE @cmd nvarchar(max) = N''
SELECT @@SERVERNAME; --this would be where your query goes.
'';
EXEC (@cmd) AT linked_server_2;
';
EXEC (@cmd) AT linked_server_1;

Essentially, the above code executes a dynamic SQL string over linked_server_1, but the dynamic SQL is actually an embedded dynamic SQL string, which executes a query at linked_server_2.