eToro does not allow transfers of any kind.
To direct register your shares, you are left with a few options:
Stop buying with your non-transferring broker and instead buy with a broker that can direct register shares. For example: Fidelity or Interactive Brokers
Sell your shares from your non-transferring broker and wait for the cash. to settle back into your bank account. Then repurchase your shares using that cash (risky due to volatility).
Buy shares in a broker that can direct register. Then sell from your non- transferring broker at a similar price, (requires extra cash, but reduces risk).
This isn't advocating day trading in any way shape or form; GME is too volatile. While you may make some gains, selling in any shape provides short hedge funds with liquidity. Those shares, and that liquidity is removed permanently if the re-purchased shares are directly registered.
Option 1: Hold and buy elsewhere
Continue to hold your shares with your non-transferring broker.
Open an account with a broker that can direct register shares. IBKR is a popular choice as it's an international broker that can direct register quickly, and for a $5(US) fee, (guide here).
Purchase a share through Give A Share to open your Computershare account (link here), then purchase more shares directly through Computershare, (guide here).
Option 2: Sell and repurchase
Please note: selling at a profit, or from a tax advantaged account may create a taxable event.
Sell a batch of shares from your non-transferring broker (or all of them)and wait for the cash to settle and withdraw it to your bank account.
Fund your new broker account with the cash now it's available.
Buy your shares with the cash. Hopefully the same amount of shares if the price is close to when you sold.
A few things to bear in mind with this option:
You may be adversely impacted by volatility. You may benefit if there is a dip, but you might lose out if the price increases.
You may be adversely impacted by transaction fees. If your broker charges for each sale, then you increase the number of fees to pay.
It's also worth bearing in mind that there will be currency conversion fees and (depending on your broker) a DRS transfer fee.
Option 3: Buy first, then sell
Please note: selling at a profit, or from a tax advantaged account may create a taxable event.
Decide how many shares you can purchase at a time.
Buy a batch of shares from the broker that can direct register. IBKR is a popular choice because they are international, can direct register shares quickly, and for a $5(US) fee, (guide here).
Go to your non-transferring broker and sell the shares for a similar price (preferably for a little more to cover the various small fees).
Wait for the cash from your sales to settle and withdraw it as soon as it does.
Wait 3 days for your new shares to settle, then direct register them.
Once the cash from your sale is back in your bank account, you can fund the broker again and repeat from the first step.
If you go for option 2 or 3, or a combination of them you do risk losing some of your investment through volatility. This would be the case for any stock, let alone one as volatile as GME. It is stressful and takes patience.