In that case, your challenges are more on the DIY spectrum than the electrical.
If cable chasing is fair game, I'd let a new single gang back box into the wall right next to the existing master, and run two category 5e solid core cables from there to a single gang back box at the far end. At the master end, fit a modular faceplate with one RJ45 module and one blank. At the remote end, you want a modular faceplate with one RJ45 module and one BT secondary module.
One of the cables goes from one RJ45 module to the other. Leave the modem where it is, and connect its LAN (Ethernet) socket to the RJ45 module using a patch cable. At the other end, connect to the router's WAN port using a patch cable.
The other cable connects to the filtered phone terminals on the back of the BT master's faceplate, and runs to the BT secondary module at the other end. Obviously, you'll need to get this cable into the back of the BT master somehow, which is why you put the new faceplate alongside the master.
In this scenario, everything that can be hard wired is hard wired, and there's no possibility of interfering with the VDSL2 signal to the modem by running it through the house. All you will see is a cable into the modem from the master, a cable from the modem into the faceplate next to the master, and the cables to the router and phone at their new location.
You'd probably save yourself a lot of hassle by getting an electrician to chase the cables for you. Not only is an electrician likely to have a powered cable chaser (typically an SDS hammer drill with a set of chasing chisels) which makes this part of the job easy, they'll have the tools and expertise to ensure they're not about to chase through existing wiring or water pipes.
It's up to you whether you wire this up yourself or get the electrician to do it - if you provide all the parts, wiring it up is a five minute job for the electrician, which would save you from buying the tools. However, as data and phone cabling is non-notifiable work, it's up to you if you want to DIY.
The making good and redecorating is probably the most significant part of the job, but should be well within the skill set of a competent DIYer.