Delegating subdomain to another nameserver...

So I will start by saying, I am not a network engineer, simply a computer scientist, programmer, fix-it guy. So no need to dumb down any answers, but keeping them within the realm of someone who knows a great deal about networking, but is not a professional by trade would be appreciated, thanks! I have also spent hours trying to troubleshoot on my own and am very close to the solution but seem to be missing one critical piece.

So here is my question. I have a site, lets call it myweb.tech. I have it hosted on register.com, but recently moved into new office space and would like to forward some internet traffic, based on the third level domain to it. So for example, I've set up a set of virtual and physical servers to handle various web applications. So far I have only deployed Windows Server 2016 as an OS on these machines (with ESXi 6.5 as a hypervisor, but that's unimportant.) So I have set up a domain controller with a DNS, another for redundancy, a web (http) server and and mail server- all on separate guest OS's. Using best practices for naming, I have named my domain zone01.myweb.tech.

What I want is to forward DNS requests for anything *.zone01.mysite.tech to the local DNSs in the domain, domain-ctlr01 and domain-ctlr02. The web server is a member of the domain, has the name web01 with an associated ptr. @.zone01.mysite.tech and zone01.mysite.tech have both been forwarded to the static IP the domain is on and port 80 and 53 are both forwarded to it. So if an external computer was to access web01.zone01.mysite.tech, what I would like is for the register.com NS to point to the main zone01.mysite.tech NS (domain-ctlr01.zone01.mysite.tech), find web01 in that DNS and point the user to the appropriate page. I can forward web01.zone01.mysite.tech to the static IP and the domain controller forwards the request properly, but not unless the complete fourth level domain is forwarded- in that case a nslookup says "the domain does not exist." I want to avoid having to put forwarders in for every fourth level domain, especially as the network grows- plus I'd like to learn how to do it the correct way.

I even added an entry in register.com's panel to add domain-ctlr01.zone01.mysite.tech the NS for zone01.mysite.tech, but had no luck.

So basically the query would go as follows: USER'S CHOSEN NS->.TECH AUTHORITATIVE NS->REGISTER.COM (.TECH) NS->FORWADED->MYSITE'S ZONE01 (3rd level authoritative) NS)->FOURTH LEVEL DESTINATION.

Is this possible? I'm sure it must be. I know there must be something I'm missing. I usually try to troubleshoot until I find the solution so I learn more, but this one has me stumped. Any help would be greatly appreciated, thanks.