Getting a domain name may not be enough, if you don’t have a static IP you’ll still need a DDNS service.
What do you get for the paid no-ip service? Is it just a nice subdomain? You can get a custom domain and use a CNAME record to point one or more subdomains to a free DDNS subdomain.
I see an option to allow time sensitive notifications for apps not in my list of allowed apps, it shows up once the “allow notifications from” option is selected. I have no idea if it actually works though, as I’ve never used it.