The point of purchasing a registered domain name and connecting it to a public DNS server is to make it findable from any Internet location. If you only ever want to use the domain name internally, you don’t need to have a public domain name and you can make up your own internal domain name that is served by your local DNS. To avoid future conflicts with public domains, I’d probably use a TLD that doesn’t exist (e.g. not .com or the like).
The point of purchasing a registered domain name and connecting it to a public DNS server is to make it findable from any Internet location. If you only ever want to use the domain name internally, you don’t need to have a public domain name and you can make up your own internal domain name that is served by your local DNS. To avoid future conflicts with public domains, I’d probably use a TLD that doesn’t exist (e.g. not .com or the like).