As your SQL Server is running in mixed mode and you want to login from some other domain, you should stay with the SQL Server authentication mode approach.
You are right to hesitate using "sa". Create a user for your application in SQL Server (get help from a database admin if you are not allowed to do so).
Grant this user not more than the rights needed by your application.
You may generate the required sql commands with a script
IF NOT EXISTS (SELECT * FROM master.dbo.syslogins WHERE loginname = N'yourUserName')
BEGIN
CREATE LOGIN [yourUserName] WITH PASSWORD=N'yourPassword'
,DEFAULT_DATABASE=[yourDataBaseName]
,CHECK_EXPIRATION=OFF
,CHECK_POLICY=OFF
END
IF NOT EXISTS (
SELECT *
FROM dbo.sysusers
WHERE NAME = N'yourUserName'
AND uid < 16382
)
EXEC sp_grantdbaccess N'yourUserName' ,N'yourUserName'
SELECT 'GRANT SELECT,INSERT,UPDATE,DELETE ON ' + a.table_name + ' TO yourUserName'
FROM (
SELECT DISTINCT table_name
FROM information_schema.columns
WHERE table_name LIKE '%yourSearchToken%'
) a