On a double mastectomy pt, we always do blood pressures on the calf of the leg.
In our institution, almost all breast cancer chemo pts have a port, so we are able to do blood draws through the port. In the case of a double mastectomy, I can't imagine the pt not having a port or some other central venous line to administer chemo, since the drug regimen is likely to include a vesicant (Adriamycin). Even Taxol can be nasty (was considered a vesicant when it first came out).
In the case of a single mastectomy pt, whose "good" arm has lousy veins and there is no central line.....we do see pts like this and it is not an optimal situation. Hydration does help, as does using some form of heat (heating pad, warm water, etc) to dialte the veins prior to cannulation. We do not use lidocaine routinely for IV starts - we use 24g needles for chemo, so their size is not so large that the pain is overwhelming. Certainly use a butterfly for the blood draw, but not for the chemo as again, it will probably be a vesicant drug and using a butterfly is not recommended for vesicant administration.
Also, I would always try to draw the blood while starting the IV - not only to minimize sticks for the pt, but to be certain the vein you are using is fresh, without other sticks possibly superior to the one being used for chemo.
FYI - I have seen a breast cancer pt get lymphedema 12 years out from mastectomy, so I never assume there is a safe time to use the affected arm. All the best - Barb