There was no basis on the information supplied to the Society to suggest the allegation should not have been made. Clearly without a response you would be unable to advance the matter on your own client's behalf. All you can do is inform the client that you have been unable to obtain a response. You should not engage in communication about the substantive issues in the matter. Nevertheless your letter will probably cause significant embarrassment to your colleague.
Some solicitors take exception to the threat but sometimes it is justified.
By this time that solicitor has been treated discourteously and might be tempted to think that his or her letters or telephone messages have indeed gone into that big black hole. Of course exception is rightly taken if it is foreshadowed at a time before a response could reasonably be expected or before the reasonable time for response had expired. As a matter of courtesy and as a matter of common sense to take the heat off yourself, you should acknowledge correspondence as soon as possible even if you have no substantive answer to it. The life of the average solicitor is stressful enough without imposing an unnecessary sense of urgency. Of course, matters become urgent because of events beyond one's control and one might have to react urgently or request a colleague to react urgently.