I have never tried to drill right through a MT arbor. However, I have cut off the tangs from a couple of drill chuck Morse Taper arbors and drilled a hole about 1/5 of the taper length without any bother to allow greater stroke of the tailstock in my 7 X12 mini lathe, which also increases the available bed length. The hole accepts the end of the tailstock leadscrew which is self extracting. It is common for soft end MT arbors to have a drawbar end rather than a tang, so I recon it would be possible to drill right through.
Your 10mm hole would only leave 2.25mm wall thickness at the small end of the MY2.