Yes, doing this at all requires the ability to create an NS record (delegate a subdomain to another nameserver). Or otherwise using a second domain name entirely.
If you are willing to move DNS hosts, you may as well choose one that supports automation and avoid this problem to begin with. Be aware that Namecheap only gives you API access if you spend a certain amount of money.
There's a list of the DNS hosts that Certbot has first-party support for which you might find helpful. Also DNS providers who easily integrate with Let's Encrypt DNS validation.
I personally recommend a Cloudflare account for free DNS hosting, it's fast, reliable and straightforward to use with Certbot. But there are other good choices as well.