Bamber may have concluded that calls could be traced on the basis the IRA always used pre-selected phone boxes to give out bomb warnings. But this was the IRA calling the police and the only way then of tracing the caller was identifying the number/line whilst the IRA were on the phone.
Coroner Sir Peter Thornton QC asked Mr Conway if the the faulty phone box story could have been a "well-orchestrated and convenient lie" for the bomb team to tell their superiors and escape punishment.
https://www.bbc.co.uk/news/uk-england-birmingham-47655204
The IRA always used phone boxes and never called on residential lines from safe houses.
Bamber was probably bluffing when he said he thought the police could check calls. Mr Bamber may have known given he was a magistrate and Bamber may have sounded him out.
'Tis a very good point this.
We have all seen the movies when someone has to keep the perp on the line long enough for the peelers to trace the call.
The fact that BT couldn't provide that information after the event was likely not such specialist knowledge in 1985 as we might assume and Bamber was just bluffing.
I do see Adam's point that an AM gives JB a way of connecting a call to Bourtree cottage just in case but this would tie him to a tight timeframe, ie he couldn't call his AM at 2.30 and then the police an hour later when he has cleaned up. It was bad enough he took ten minutes to look in a phonebook which even in 1985 were listed in alphabetical order!
I vote bluffing.