Definitely, hold the Die in a holder that slides along an arbor in the Tailstock.
Asking a M4 thread to drag a Tailstock along the lathe bed is asking for stripped or damaged threads.
Definitely lubricate before starting the cut, Trefolex or ROCOL RTD.
And put a chamfer on the end of the work, to help the Die centre itself, and start to cut.
Since this is Silver steel, it would be worth taking the cut in two stages. Open up the Die, as far as possible, without cracking it, and take the first cut.
Then for the finishing cut, reset the die, using an M4 fastener as a gauge.
This will reduce the cutting loads, and the risk of problems.
Ideally, with hard material, rotate the chuck by hand (Mandrel handle, or using the chuck key; minimal risk of danage to chuck key, since this is a small thread)
HTH
Howard