I guess you should export your sequence with Quicktime Conversion, select avi and choose the right compression settings (probably None).
As for the 4 GB limit, you can get around that by defining a Limit to the Export File Segment Size in the System Settings (Scratch Disk tab). FCP will chop up your export accordingly into smaller files that will open as one movie on the other side.
I am not sure though if this all works out nicely on the Premiere side of the universe – I regularly did this with .movs that had to be transported to a PC and that went flawless, but I have no experience with avi’s. (By the way if your Premiere studio has Quicktime installed they should be able to open a mov)
Jaap