Not nice!
If I have understood correctly it is hard to see how mode s would have helped, unless you have TCAS on your aircraft?
“The incident occurred in the few moments between changing service providers”
You were changing to a new LARS unit so not actuality getting a service? I assume you have mode C and that the other aircraft was doing 160kn + it probably had c or s. Mode C or Mode S are excellent ways of avoiding collision if the aircraft concerned are TCAS equipped, but of little value otherwise.
I had the following incident some years ago.
Tracking to a VOR in quite solid IMC, on a RIS. I had traffic reported 1000 ft below unverified. I had class A above me, so stayed at my height with a theoretical 1000 ft of clearance. Popped out of the side of a cloud to see the other aircraft flash under me at about 50 ft separation. The controller confirmed it was the reported contact, still indicating 1000 ft below. The encoder must have been out by 950 ft! In this instance even TCAS would have been of no use. I “modified” my approach to flying IFR after this, I now like to see what is coming the other way.
Rod1