You need to reduce the size of the log file first. If Full recovery model is truly needed in your environment then you need to rethink how often you are doing transaction log backups in order to keep the log size manageable. You can do transaction log backups as often as every minute if need be. If you don't need point in time recovery, think about switching to a simple recovery model.
First things first, backup the transaction log and then shrink it down to a manageable size for server 2, you may have to repeat this step a few times in order for the log file to reduce in size. Once you have reduced the transaction log size to a reasonable number, take a fresh full backup and use that for your restore on server 2. I would definitely recommend a more frequent transaction log backup schedule on server 2 to keep the log file under your 300 GB hard-cap.
To see the transaction log backups run this script
SELECT user_name, database_name, backup_start_date, backup_finish_date, backup_size,
WHERE type = 'L' AND database_name = 'YOURDATABASE'
This will give you backup start, backup finish, size of the backup, and compressed size if applicable. If this field is blank and you are in full recovery model, whoops!