How do I get the CNAME record to verify with Mailgun, so emails can be sent via this new sending subdomain?
Cloudflare’s default universal SSL certificate only supports root and 1st level subdomains. If your tracking hostname is a 2nd level domain or higher, you’ll need to contact Cloudflare to discuss your options for an SSL certificate to support the tracking hostname.
This only applies if you have proxy enabled for that record, you don’t, so none of Cloudflare’s SSL stuff matters.
I would walk through their tutorial https://help.mailgun.com/hc/en-us/articles/15585722150299#01H03Q1V4V8F5GNJ072J3S0YJA and double check the name. From that guide, it seems it defaults to email.mg, not email.fgm, not sure if that’s something you changed.
dig +short mailgun.org
dig +short email.fgm.yourdomain.com
It looks like you have CNAME-flattening activated for all CNAMEs. Also, you didn’t obscure all occurences of your domain in your screenshot