Thank you for your post into the forum.
With regards to filegroup backups and the restore process, is exactly the same as using native SQL Server backups. So to answer your questions:
I would like to be able to do a one-time full backup (all filegroups) and after that I just want to backup the dynamic portion. Is this possible?
The answer is yes, if this the course of action you wish to take.
I case of catastrophic failure, what would the restore process entail?
Restore the Full backup, next the latest filegroup backup and followed by the unbroken chain of transaction log backups from the original Full Backup.
Whilst you wish to take a one time only Full Backup, you may want to reconsider this action and perform a regular full backup to reduce the number transaction log backups for restore. Also taking regular Full backups, if the one time only full backup file becomes corrupt as the file maybe moved around storage locations. You wil be unable to recovery the database if a catastrophic failure occurs.
Also perform and test your recovery strategy to ensure that it meets your needs and that of your organisation.