Currently, how are your clients setup? What are their www and root records pointed to?
For load balancing, all you need to do is CNAME your customer to your firewall/load balancer. So you aren't using A records for this. For example, in Azure, if you spin up a traffic manager, you would get an cname like "mytrafficmanager.trafficmanager.com" and your CNAMe for www.mysite.com would point to mytrafficmanager.trafficmanager.com.
However, in this case, you would also want your customers to point to something like customer.mysite.com so that if you move from GCP/Azure to something else, you can handle that record and migrate them during a failover, incident, or any other reason.
Edit: And have customer.mysite.com point to the "mytrafficmanager.trafficmanager.com"
True, spinning up nginx and setting that up is the cheapest option I have come across with the best enterprise support. It's also available in Microsoft to deploy instead of using their tools. However, some people like being cloud native.
For load balancing, all you need to do is CNAME your customer to your firewall/load balancer. So you aren't using A records for this. For example, in Azure, if you spin up a traffic manager, you would get an cname like "mytrafficmanager.trafficmanager.com" and your CNAMe for www.mysite.com would point to mytrafficmanager.trafficmanager.com.
However, in this case, you would also want your customers to point to something like customer.mysite.com so that if you move from GCP/Azure to something else, you can handle that record and migrate them during a failover, incident, or any other reason.
Edit: And have customer.mysite.com point to the "mytrafficmanager.trafficmanager.com"