Accepted answer
Write the units at every step, because units errors are the failure mode that catches everyone eventually.
Room temperature before drawing is worth the ten minutes. Cold solution is more viscous, draws slower, and condensation on a cold barrel makes it harder to read the meniscus.
Concretely, breaking it down further: if a 10 mg vial has 96.5 per cent content, you have 9.65 mg of peptide. Divide that by 2.00 mL and your concentration is 4.825 mg/mL, not 5.00 mg/mL, which is a 3.5 per cent systematic error in every dose calculation.
The Arrhenius relationship for drawing kinetics means that cold solution takes noticeably longer to draw than room-temperature solution.
If in doubt, use more diluent and accept the shorter usable window.