Why does "Transferring Purchases" from iPad in a USB Backup Fail due to lack of space?

I'm having a terrible time trying to backup my iPad ever since I installed a large game yesterday. I am stuck with a 16GB iPad that has given me all sorts of trouble from day 1 due to lack of space.

I had more than enough space on the iPad to install the game (~3GB) when it was purchased.

Now, when I connect the iPad to my PC to back it up through iTunes, about 30% of the way through "Transferring Purchases" of that game, I get a message that the iPad cannot be backed up due to lack of space.

Is there a way to just transfer the purchase, outside of a full sync or backup?

I've resisted using iCloud backup because Apple services have not been reliable in the past and because I don't have nearly enough space in iCloud to back up 16GB of data. Would iCloud backup work?

(If it's not clear from the headline, the lack of space is on the iPad. My PC has tons of space.)