How are you holding the die?
Although it is possible to use the tailstock as a makeshift die-holder guide / pressure pad (I have done it), it is not very reliable for very accurate work (I know – I've tried it).
Your second best bet to screw-cutting is to use a tailstock die-holder, and that will probably give you as good a result as you are likely to expect or need.
Is your difficulty compounded by expectation? You seem to mean this is for a piston simply screwed to its rod. Whilst I don't know the specific engine, it is usual to not rely on the thread alone for concentricity. Instead, the assembly should include a register, at its simplest a plain section of the rod fitting a reamed portion of the hole in the piston.
Many builders also leave the piston a tiny bit over-size then skim the last few thou off with it finally assembled to the rod, held between-centres or by other means to ensure concentricity.
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Can you cut BA threads? Well, they are metric though not obviously so; to a geometrical progression.
5BA is of 0.59mm pitch, or 43 TPI., and 0.014" (0.355mm) deep.
If you screw-cut it about 0.7 thread depth, you can die-cut it to size and profile by die, and on a thread of less than perhaps 8 or 10 turns the pitch error is likely to disappear in the cutting tolerance. Anyway, a slightly tight thread is probably an advantage on a piston-rod.
The awkward bit is grinding the tool accurately without a basic tool-&-cutter grinder.
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If your lathe is to Imperial dimensions it is possible, using spread-sheets, to find combinations of its change-wheels that will theoretically generate some close-match mm / BA threads for short, though useful, distances before the accumulating error prevents correct finishing with a guided die.
Edited By Nigel Graham 2 on 13/01/2021 00:42:16