SQL Server Timeout exception for multiple Users using SQL Stress
The timeout is with the connection pool, which is a collection of connections maintained by the client.  All of the connections in the pool are in use, and it is timing out getting a connection object from the connection pool, not timing out trying to connect to the SQL Server.  For example, you may have 30 connections in the connection pool, and a 5 second timeout for obtaining a connection from the pool.  A connection pool timeout will occur if all 30 connections in the pool are busy and none are freed up within 5 seconds of a request for a connection pool object.

So you can increase the number of connections in the pool, increase the connection pool timeout, or work to see if the stored procedure execution time can be improved.  It is likely that all of the connections are busy due to blocking, and if that's the case, increasing the connections in the pool and the timeout won't increase the number of times the stored procedure can execute per minute and you'll need to find out how to make the stored procedure execute faster.  So a good first step would be to look at the execution plans and missing indexes for all of the queries in the stored procedure.

Perhaps obvious, but since you state it works better on the "local" server, what are the configuration differences between the two servers?  If read committed snapshot is enabled on the local server but not the other, that one config change could have a huge impact.