Associate Elastic IP to ELB network interface

That is (unfortunately for your use case) how Application Load Balancers and Classic Load Balancers behave. You can't assigned a fixed IP to them.

One solution is (as you say) to use a Network Load Balancer in front of the load balancer you're using. I'm not sure how this will impact your business applications but let's assume that it isn't a viable solution for you.

You've hinted that the issue here is that you can't use an A record with an Application or Classic Load Balancer. You can - but your domain needs to be hosted using Route 53. So consider that as an option.

The other way to do this is to have something (Lambda perhaps?) monitor the DNS records for your load balancer and update your DNS servers with any changes. This isn't ideal but in general, when a load balancer is taken out of service (for scaling or lifecycle purposes) it isn't immediately terminated. It's still there while existing connections drain and while DNS records that hold the IP address for that node age out for whoever has made those queries. So you do have a little time to make this change without impacting your clients.

It's a bit more work to do it that way (using Route 53 is cleaner) but if you can't do that then I'd suggest using a script of some sort.