I just replayed the game, and I figured out exactly how trading works, so
When trading, you HAVE to make sure to use another name/deck. If you use the same name [profile] on both memory cards, it won't matter that you're using 2 different memory cards. [Like, if you called yourself "???" on both memory cards and tried trading in "DECK" mode, it won't let you. But if the names are different, it will. So have different names. That's how you trade.
So this means you CAN trade in the same memory card, but it might get a bit confusing. Just make sure your 2 profiles have different names.
You can't trade cards that are on your deck, but you CAN take cards off your "Main Deck" [but you have to keep 3 monster cards on any deck at all times, didn't check how many arena cards you need to keep] by doing this:
1) Go to DECK mode from the main menu 2) Load the save profile that has a card you want to take a card off its battle deck to trade 3) Go to MAIN DECK - You should see cards that aren't on your battle deck. 4) Click L2 or R2, this will bring you to your "MAIN DECK" where your monster cards are. Like, for me, I used SHODAIGOJIRA in my playthrough, so he is in there. But I never used Ganimes but got a card of him, so he is not in there. 5) Click X on the cards you want off your MAIN DECK. You can also go to the right or left to take off arena cards [and battle cards and attack cards]. - If you have duplicate [multiple] cards, you can cycle through them by pressing the L1 button. Check the right to see what level and whatnot the card is to see which is the one you want to take off. 6) Press L2 or R2 again to go back to your "STOCK DECK" [where all the cards that you don't use in battles are].
Go back to the DECK menu, then save your changes. Then go to "TRADE." Select profile #1, and then select profile #2. After, you'll appear in the screen with all your cards. Select the card you want to trade with X, and select another card from your other profile. You can cycle through cards with L1 if there's 2 cards of something and one of them is in your deck while the other isn't.
After, it will ask you if you're sure you want to trade, so put "YES" and then save it to the exact profiles you put before.
So, just like Pokémon trading, trading a single card can take up to 2 minutes.
And that's how you trade! I think what made this impossible was that no one could read the Japanese. I translated the little "error" thing it gave me when I tried trading between two profiles [with the same name] and it said "Can't trade between the same person," so I started a new profile with a different name and that worked.
So I hope this was helpful! I think I might end up making a video on "How to play GODZILLA: TRADING BATTLE" because of this, in addition to adding these newly-found instructions to the Wikizilla page.