Backup connectivity design

Hi, I have 2 connections at home. Primary is using static public IP, so no issue to link this with Next DNS. However, backup is using NAT, no public IP and even dynamic one. 

In case of a failover, IP will change and Next DNS servers will stop resolving. Is that correct assumption? I have tested to use the Next DNS servers IP on anonymous device and resolving is not working at all. I would be completely fine getting resolving without the filtration (it is backup after all), so the same scenario as if I exceed 300 000 lookups, but this seems not to be the case. 

Any ideas, how to workaround this? Using DynDNS does not seems to be great for such critical service as DNS. 

Thanks for any feedback

Martin