Two Databases on same server handle date differently
If I understand the problem correctly, when both users have British as default language, one user's request goes through fine because the language setting changes automatically to us_english.  But for another user, the language setting doesn't change automatically, hence the error.  

The BOL states following regarding default language and how it can be overridden.  


  The default language for a login can be overridden by using CREATE
  LOGIN or ALTER LOGIN. The default language for a session is the
  language for that session's login, unless overridden on a per-session
  basis by using the Open Database Connectivity (ODBC) or OLE DB APIs.
  Note that you can only set the default language option to a language
  ID defined in sys.syslanguages (0-32). When you are using contained
  databases, a default language can be set for a database by using
  CREATE DATABASE or ALTER DATABASE, and for contained database users by
  using CREATE USER or ALTER USER. Setting default languages in a
  contained database accepts langid value, the language name, or a
  language alias as listed in sys.syslanguages.


Based on this the following can affect the language setting.


Database default
User default
DSN setting, connection strings, etc..
SET LANGUAGE in session


If underlying statements can't be changed, is there a setting that can affect it in the third party app?  Or is there are trigger for a certain user login?  Though I believe it's one or a combination of 2 or more in the above list....