Backing up MS SQL Server database creates a huge transation log
The statement "transaction log is getting properly truncated on the full backup but it immediately fills up after the backup" is not how backups operate though.  All transaction log truncation is done during the first transaction log backup after the full backup, the truncates are deferred until the full backup is completed.

What happens to transaction log backups during full backups?

If you are seeing a bigger transaction log after a full backup but before the first transaction log backup, that's typically by design.  However if you are saying you have almost no activity going on during the full backup which would cause a gig of activity then I would immediately start by looking at your jobs.  Often a reindex will cause a lot of logging but it would be odd for that to be the main culprit.  Carefully monitor what happens during that period and run the same queries.  Feel free to profile what is going on during that time as this could be a compound issue.