This next engineer might be more persistent, I think they have an obligation to delve on second visits ? Others on here might clarify.
No engineer worth his salt wants to attract a repeat report (repeat visit), nor would the business, for obvious reasons.
It not only affects the customer experience, it also gets logged on the engineers stats and is costly to the business as an extra 'truck roll'. So a 2nd visit should see deeper scrutiny applied, when fault-finding.
There is of course the elusive intermittent fault that disappears the minute the engineer knocks on the door, but a 'bridged tap' result doesn't fall under that category, it is a hard and fast fault condition.
I admit, i'm struggling to get my head around your internal set-up, but would mention the engineer is only responsible up to the master socket. The 'proving' of the bridged tap fault is relatively straightforward, as the engineer should run a GEA test on his mobile phone that is pretty much the same test as your ISP runs.
If when he arrives at your premises it tests bridged tap in extension 1 socket (or however it's wired), he/she should then go to the master socket and re-run the test ... if it's clear at that point you know the fault is inside. If it were me, I would also perform the test with my own Hand Held Tester (HHT) connected, not your own router.
I have to say, in all the years I've performed BB faulting - I've only ever had one bridged tap fault outside - and that was somebody accidentally terminating a circuit they were working on, over the pair of wires that I subsequently picked up as a fault. This was done in the Cab, and on the Quante strip connections.
As in all walks of life and all careers, there are those just starting out on their knowledge path and those at the end of it. Most engineers I've met are happy to chat with the customer and explain what they are doing regarding testing, but not all will be aware of the little tips & tricks that can be used to help identify faults.
Maybe start a convo with whoever rocks up over a cup of tea and take it from there ??