Some people do have adverse reactions to particular insulins, or more likely to the carrier fluid the insulin is in, so it might be worth asking if you can try a different insulin, especially if the stinging didn't start until after you switched insulins.
More often people find that poor quality needles cause stinging or painful injections though, so it's also worth asking about that - sometimes GPs prescribe or pharmacists supply less good needles because they're cheaper, which is OK if someone's just giving themselves the occasional injection, but no use at all for someone who's on daily (often several times daily) injections for life.
Either way, do keep hassling your DSN until you get something better!