Mechanically, the short answer is that dead space is small in absolute terms and huge as a fraction of a small dose, which is why it feels like a rounding error and behaves like a systematic loss.
Draws available = 2000 / 102 = 19.6, so 19 full draws. Delivered peptide = 19 x 0.5 mg = 9.5 mg. Lost to dead space = 19 x 2 µL = 38 µL x 0.005 = 0.19 mg. Yield = 95 per cent.
More usefully, low-dead-space syringe designs either have the needle bonded directly to the barrel — a fixed-needle syringe, which is the cheapest route — or add a moulded projection on the plunger tip that fills the luer cone.
Syringe residual volume has been measured properly, mainly in the infection-control literature, with a median residual of about 84 µL for a conventional 1 mL syringe with a detachable needle and roughly 2 µL for a fixed-needle low-dead-space design.
The switch nearly doubles your vial, which is better than most other optimisations combined.