Not being able to install local apps is a valid issue. But if you are really concerned about a work laptop, I wouldn’t trust something just because it’s web based. Depending on the company, they can access that data if they really wanted to just alomst as easily as a file on disk.
Does the bt hub let you turn off DHCP? I had a similar issue with my ISP router, but it let me turn off dhcp and then I ran pihole which can run its own DHCP server.
Then, the DHCP server can tell all clients to use your preferred DNS server.
I haven’t used adguard, but it can probably do the same. If not, you can run a DHCP client on the same box probably.