Accepted answer
It gives 1.67 mg/mL, and whether that is sensible depends on the dose you will draw from it. 5 ÷ 3 = 1.67 mg/mL in bacteriostatic water. A 0.5 mg dose is then 30 units on a U-100 barrel and a 1 mg dose is 60 units. Both land high on a 0.3 mL barrel; a 0.5 mL barrel or less diluent would be tidier.
Stated carefully, this is the one decision in the whole preparation sequence that cannot be revised later, which is why it is worth thirty seconds of arithmetic.
Round to a diluent volume you can measure accurately. Measuring 1.00 mL on a 1 mL syringe is reliable; measuring 1.37 mL on anything is not, and the error propagates into every dose.
Vial headspace is the hard constraint. A nominal 2 mL vial typically holds a little over 2 mL to the shoulder; adding 3 mL is not an option and attempting it wastes the lot.
Insulin syringe barrel graduations are typically 1 unit on a 0.3 mL barrel, 1 unit on a 0.5 mL barrel and 2 units on a 1 mL barrel, which is why the barrel size changes what is readable.
Do not change the diluent volume between vials of a titration without recalculating; it is the commonest source of a ten-fold error.
Check the vial can physically hold the volume before you draw it up.