Are you having some problems using ArcMap 10.3.1. with Runtime .NET? I believe that you should be able to use any version of ArcMap 10.1.1 or higher to create runtime enabled map packages and 10.2.X or higher to create runtime content.
Could you clarify what you are doing with Runtime and ArcMap? If you are talking about using ArcGIS Runtime for .NET to extend ArcMap then that is not supported. For that you need to use ArcObjects. If you are using ArcMap to create data/services/geoprosessing packages and have problems with that could you tell more what you are doing?
There is a known issue creating map packages in ArcMap 10.3.1:
Map package created with runtime support in ArcMap 10.3 shows broken link when 'Reference all data' is checked
There is a hotfix for that but I don't know if it was made public. The problem is marked as fixed for 10.4.