SQL Server - Restore only primary Filegroup but bring the database online
I know you mentioned it's a little bit out of your control, but if you don't need the data at all and just the schema itself, doing a backup and restore on a 2 TB database is going to be the least efficient way to accomplish your goal, by far, among probably 10 different options. (You should familiarize yourself with the alternatives and bring this to the attention of whoever's control it is in.)
Some of those alternative options are Replication (as you mentioned), SSIS, SQL Agent Job that keeps the other instances up to date, scripting out the database, or using a DACPAC to deploy just the schema, which would be my choice for you as it's the simplest / makes the most sense. (Also if your limitation in using a different features is because you're not allowed to setup anything on the server, DACPAC is an already enabled feature, and the process is simpler than backups since you just use SSMS to generate the file for you, similar to if you were generating scripts. No need to install or setup anything.)
If your schema doesn't change often, you can even dedicate one empty copy of the database as your &quot;model&quot; which always remains data-less and can easily be used to generate new databases off of (even if you still stick to the backup and restore methodology, since it'll be a tiny database). Or even if it does change often, you can still do the backup and restore to just this one model database, and then truncate the data or use a DACPAC there to generate the other instances so that way you don't need to wait on a 2 TB restore more than once.