I personally would react as follows;
any RA will be carried out in full nomatter what.
A non mode C equipped A/C would never generate a RA but a TA and the target is identified by the VDU as a non mode-c target. I would then ask for ATC confirmation and if non is given or can be given, then I would reason that the non-C a/c is below radar coverage and therefore no threat to me at my cruising level. However, in IMC low-level one could reason that the non mode-c A/C is VMC and therefore also no threat. And if low-level VMC I would keep constant lookout. There is no reason to change course nor altitude because of a TA on a non mode-c target as the altitude change may lead to disaster just as much as maintaining altitude. Course change can be considered, but using the VDU for guidance is totally useless as the directional error on these are as much as +- 60 degrees in some atmospherical conditions. If ATC has radar contact with you and not the a/c that caused the TA then your altitudes are very likely not similar, and should be no threat.
To this extent, I feel safety is well preserved and would not have any problems continuing VMC or IMC..