Right, I'm clearer now, and also hadn't answered a couple of your questions.
With FTTC, which is what you are getting, the fibre optic stops at the cabinet. Hence the name. From there to your master socket is the existing copper. There is no fibre optic reaching the house.
I assumed you get to keep the HH2. All I described was using it as a WAP, rather than buying one. Quite often done when someone has two routers and inadequate wireless coverage. But I didn't suggest the best layout.
What you need then is your pre-installed CAT 7, preferably two cables but one will work, running between the existing master and the extension. Leave some spare for an engineer to work with.
You get the engineer to move the master to where the extension is, using that CAT 7. They do
move the master if they are helpful. Then you want the extension installing where the master is.
There have been many people on these forums where the master has been swapped with the extension, or simply moved. The engineer will not install the modem more than a short A/VDSL cable distance from the master.
Setup then is the incoming VDSL2 supply goes to the master in the study, with the HH3, and the PC wired to the HH3. The second CAT7, or spare pairs in the only CAT 7 if that's what you install, (apparently this isn't ideal in theory), takes a phone connection and an ethernet feed back to the HH2 where it is now, where you fit an ethernet socket alongside the extension socket.
Disable DHCP on the HH2, play around with its 192.168.etc IP address to keep it outside the HH3's DHCP pool, and you should be fine.
The interstitial plate that fits the NTE5 master to provide the VDSL filtering and modem socket does apparently have an unfiltered output if required, but that wouldn't solve your problem.
Combined modem/routers do exist, notably the Fritz box seems popular here, but is not Openreach approved. In particular if you have a problem you would need to be using the modem and HH3 to get anywhere with support.
My broadband basic info/help site - www.robertos.me.uk
Domains,website and mail hosting - Tsohost
. Connection - Plusnet
Extra Fibre (FTTC). Sync ~ 56.0/13.9Mbps @ 600m.
"Where talent is a dwarf, self-esteem is a giant."
- Jean-Antoine Petit-Senn.
Allergy information: This post was manufactured in an environment where nuts are present. It may include traces of understatement, litotes and humour.