Acme dns challenge works for things that aren't webservers.
For the other case perhaps renew the cert at a host allowed to do outside queries for the dns challenge and find some acceptable automated way to propagate an updated cert to the host that isn't allowed outside queries.
Last time I checked there's no standardized API/protocol to deal with populating the required TXT records on the DNS side. This is all fine if you've out-sourced your DNS services to one of the big players with a supported API but if you're running your own DNS services then doing automation against that is likely not going to be so easy!
One pretty easy way to do it while running your own DNS is to put the zone files, or some input that you can build to zone files, in version control.
There are lots of systems that allow you to set rules for what is required to merge a PR, so if you want "the tests pass, it's a TXT record, the author is whitelisted to change that record" or something, it's very achievable
I was just digging into this a bit and discovered ACME supports a something called DNS alias mode (https://github.com/acmesh-official/acme.sh/wiki/DNS-alias-mo...) which allows you to add a static DNS TXT record on your core ___domain that delegates to a second ___domain. This would allow you to setup a second ___domain with DNS API (if permitted by company policy!)
Giving write access does not mean giving unrestricted write access
Also, another way (which I built in a previous compagny) is to create a simple certificate provider (API or whatever), integrated with whatever internal authentication scheme you are using, and are able to sign csr for you. A LE proxy, as you might call it
It also sounds like the right people to handle certificate issuance?
If you are not in a good position in the internal organization to control DNS, you probably shouldn't handle certificate issuance either. It makes sense to have a specific part of the organization responsible.
For the other case perhaps renew the cert at a host allowed to do outside queries for the dns challenge and find some acceptable automated way to propagate an updated cert to the host that isn't allowed outside queries.