Ordinary consumer SIMs generally don't let you arbitrarily substitute a landline CLI.

Twilio isn't a mobile network operator and can't turn a normal EE02 Vodafone 3 SIM into that kind of extension.

There are, however, specialist fixed mobile convergent slash hosted PBX SIM products

where the SIM itself is provisioned as an extension of a business telephone system.

Those are much closer to what you're imagining.

Calls arrive over the mobile network, there's no SIP soft phone to keep alive

and the business landline number can be used as the presented identity.

And I think that is actually the avenue worth investigating for you

given what you were saying yesterday about GroundWire.

GroundWire works, but what you really want is essentially, open quote,

a genuine Vodafone mobile SIM whose telephone identity is your Sheffield landline number.

Close quote.

That would eliminate the push notifications,

SIP client complication altogether.