First of all, this Apex Legends Mobile Regional Beta is currently available only in India and Philippines, to register successfully, you need to register in these countries/regions first, otherwise there are solutions for that too.
To pre-register for Apex Legends Mobile, you can simply use a VPN to manipulate the country/region and then select India and the Philippines to pre-register.
The steps for success are as follows:
1.1 Connect using the VPN of your choice but make sure you are connected to the VPN servers in India and the Philippines, otherwise, it will not work properly.
1.2 Use the VPN to create a new Google account.
1.3 After creating an account, go to the "Pre-register" link
1.4 Now, you can see the "Register" option here, so click on it and you are done.