If they get the message "we do not have the callers number" then the CLI identifier is not present in the c7 string (probably replaced with zeros) and so, no, you cannot get around it. If you get the message saying "the caller withheld their number" then it is technically possible to override the withhold, which the operators and emergency services do.
As for using a mobile, ot might work, you would have to try it and see.
alternatively, why not buy them an answering machine ?