I’ve had a similar issue arise in the past, so I’m eager to see a resolution to this question. Aaron Bertrand’s comment on the OP led to Query times out when executed from web, but super-fast when executed from SSMS, and while the question is not a duplicate, the answer may very well apply to your situation.
In essence, it sounds like SQL Server may have a corrupt cached execution plan. You’re hitting the bad plan with your web server, but SSMS lands on a different plan since there is a different setting on the ARITHABORT flag (which would otherwise have no impact on your particular query/stored proc).
See ADO.NET calling T-SQL Stored Procedure causes a SqlTimeoutException for another example, with a more complete explanation and resolution.