Can Opioids alleviate some of the painful symptoms of Benzodiazepine withdrawal? - Most certainly. I'm sure having access would be a major aid in softening the severity of your symptoms, however, unfortunately, Benzodiazepines and Opioids work upon different systems in the mind and body, meaning you can't substitue one for the other. Even if you completely anesthetize yourself with painkillers, you are still going to crave Benzodiazepines throughout the process.
I've experience with this myself. Going through alcohol withdrawal I dosed a pretty large dose of Methadone with little Opioid tolerance and became heavily sedated and felt much better, but I still really needed a drink. I wouldn't have been able to relax regardless of the amount of Opioids I had taken.
OP, what's your tolerance to Benzodiazepines at the moment? If we had a little bit more detail, I and some others would probably be able to give you a better approximation of what you might experience.