thiswoodeno you need to have enough space on your ipad to back up to your ipad once you have a backup on the ipad you tranfer it to your computer then you can delete it from the ipad, but you definitely have to have 7.3gb of Space on your ipad.
Firstly i would suggest check that you havent made an old backup or two before, cause that will chew up your memory quickly it will be in the ST3 data folder, data folder, on my ipad, ST3 folder.
As 1000 songs doasnt seem a lot it should not be hogging up 7.3 gig unless you have made multiple backups in the past or use really big audio files, and havent deleted them i had this issue with a 215 gig ipad and about 35 gb of St3 data i soon learned not to have more than one backup on my ipad.
If thats not an issue then the only thing i can suggest is have a cleanout of all things you havent used on the ipad in the last year and hope you get enough space, Sologuy has also sugested a full ipad backup to your computer, this i would also suggest no matter what as you never know what happens to your ipad.
But i also have a ST3 full backup on my computer so i can transfer the full setup of ST3 to all my other devices that have ST3 on them that way i can use them knowing they all are set up the same.
Good luck i hope this info has helped.