So - you could try manually modifying that connection string to use the credentials you want.
Remove:
Integrated Security=SSPI;
Replace with the SQL credentials you want to use:
Password=YOURPASSWORD;User ID=YOURUSERID;
Save the changes to manifest.xsf. Close it. Right click on manifest.xsf and select Design - it will open in design mode. Don't overwrite your original file, just in case something is not right, but use File > Save As to save as an XSN again. Then you can try previewing and make sure it works, or publish it and have someone test it who doesn't have access to the database.