Ideally, you would find the geometry type of the layer you are pasting into. Open the layer properties and click on the information tab. In the 'Information from provider' section, it will say what geometry type the layer is. I suspect it is PointM. You can then right click on your shapefile, save as, and set the SHPT to POINTM.
If, for whatever reason, you cannot convert your point properly, you will have to construct a new shapefile of type pointzm (or whatever type you need). To do this:
add 4 new fields - x, y, z, m
Calculate x coordinate into the x field, y coordinate into y field, z coordinate into z field, m coordinate to m field.
Here is a field calculator example for m: m($geometry)
I don't have any 25d files to work with, but I have read that QGIS has trouble accessing the z value directly. If this is the case, you will need to convert to text and extract the z value
Use the 'Create Points Layer From Table' to create you new pointzm shapefile by selecting the appropriate columns (x,y,z,m) from the shapefile you constructed.