The age old question ?
It all depends on the knowledge level of the engineer in attendance ?? However, from your description I would insist on a Broadband engineer and hopefully you can emulate the fault conditions to him when he arrives on-site. Most broadband engineer will have a level of network engineering skills, so should be able to cure what sounds like a 'High Resistance' fault on your circuit.
Conversely, network only engineers are just that, they only work up to the NTE and don't possess Broadband skills. Plus, they will just run a PQT test which may easily pass as OK dependant on the severity of the HR fault. Then they will listen on your landline with their 'Tapper' (Phone) and usually without the router plugged in. If they can't hear any audible noise, then they will leave and you may get a bill.
Of course, there's a lot of debate around this practice, but it has certainly halted the pedantic EU's chasing 0.0002 dB on their upstream SNR.
Alas, as always, there is the others who have to suffer a price.
But, it does smack of a decent HR fault you have, and as I say a BB engineer should have most, if not all, angles covered.