No. "SQL Server Authentication" only works with SQL Server inputs, it will not work with NT input. To log in as another credential, the process starts as the credentials you want to run. Use runas , most likely you want to start SSMS as follows:
runas /netonly /user:domain\user "c:\program files\...\ssms.exe"
By specifying the /netonly argument, your SSMS will start with the required NT credentials only for remote hosts, the local credentials will be local. This not only saves your SSMS environment (last used files, parameters, etc.) and does not require domain\user have local privileges, but more importantly, it works even for a completely unrelated domain . That is, if you work in the foo domain (or do not even join the domain), and the required runas are in the bar domain, runas /netonly /user:bar\user ... will work fine.
The /netonly trick only works if the host of the SQL server you are connecting to is remote (not on the same computer). If local, then your runas should be local using the required credentials, so remove the /netonly argument.
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