Two people working through the same arithmetic independently should get the same answer, and if they do not, someone has made a unit error.
Rotation of injection site is a tolerability measure, not a pharmacokinetic one, but if you are going to do it you might as well do it right.
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.
I would flag the obvious failure mode: people get the concentration right, get the volume right, and then read the syringe against the wrong scale.
Do the arithmetic twice, ideally with someone else doing it independently.