There is no style file copy on the GW8. You can LOAD new styles, but the ROM style section is not available to the file system.
Sadly, the only way to copy the ROM styles is to painstakingly make SMF's of the style divisions (not forgetting about the different patterns for maj/min/7th) and then reassemble them with your MIDI to style capabilities in your E/G-series arranger.
And even then, you are up against it... Personally, I think one of the BEST things about the GW8-L is the very good Latin Percussion in the expansion ROM. Most of these sounds don't even exist in the E/G-series arrangers, so a translated style still has quite a battle to sound as good.
I know that Roland have moved on to ARX expansion for their Fantom WS line, but there is still a ton of current and legacy products with SRX slots. I am surprised that Roland hasn't taken the Latin and European and Asian expansion ROM's for the GW8 variants, and released them as SRX cards, for adding to Sonic Cells and G70 and E80 arrangers. I can see Roland making a decent chunk of change from us if they bundled the SRX card with GW8 styles translated to work on the card/arranger hybrid.
But them again, Roland seem hell bent to destroy any chance of success in their arranger division. So this fairly obvious move is probably pretty unlikely...