Copy SQL Server DB into another instance without SA access
Is there any better way to copy the database without SA access?  


Yes, you do not have to be sa or sysadmin member to perform the activity that you described, but following are the least privileges required to do so.:

You must be member of 


Database role: db_ddladmin or Database role: db_owner at source server  
Server role: dbcreator at destination server  


You can verify current membership with following query (1=YES, 0=NO):

---- Source server
select  IS_MEMBER ('db_ddladmin' ) as Is_DDLAdmin, 
        IS_MEMBER ('db_owner' ) as Is_DBOwner
go

---- Destination server
select  IS_SRVROLEMEMBER ('dbcreator' ) as Is_DBCreator
go


I haven't used dbForge studio, but using SSMS free tool provided by Microsoft, and while having permissions (as mentioned above), you can achieve what your looking for (export data along with objects into .sql format) 

via  SSMS -> Tasks -> Generate script option. focus on following options at Advanced scripting options


Types of data to script = Data and schema  
Script for server version = your desired version