It’s best practise to have at least 2 DCs anyway, but setup a second, on the remote site, configured for each to use the other as their primary DNS, then 127.0.0.1 as their secondard.

You may want to configure sites-and-services, so users over that ‘site’ get ‘their’ DC first. Update the DHCP scope for DNS to give users on ‘remote site’ ‘remote DC’ for DNS, then ‘HQ DC’ for secondary.

As for data, you may want to look at a DFS share with replication, again, you would use ‘sites-and-services’ to send users to their closest one first.

You don’t want a child domain, this is legacy.

There is too much details to give for the complete setup, but the above should point you in the right direction for some reading.

TL:DR
Read up on sites and services
Best practises for DNS
DFS-R

Active Directory DNS Refresher - Windows - Spiceworks Community

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