Naloxone or naltrexone will help you with is to get the opioids out of your system, in particular to dislodge them from their binding sites.
Naloxone or naltrexone won't speed up your body's recovery (rather, this is highly unlikely) to a closer point of homeostasis, or recover from the shock of withdrawal. Actually, using naloxone in the way you described (unless you waited long enough for the agonists you've been using to clear your system on their own before taking any naloxone), will exacerbate the shock of withdrawal to your system. Even if you can and want to take the pain, it's not going to do anything for you or speed up your "recovery."
Unless you have a significant amount of naloxone, often OD "reversal" kits don't include all that much naloxone. I would think you'll need more than what is provided in one "kit" (esp. if you have a certain type of nasal spray/intranasal kit or one of those weird pen things). You'll have to probably repeatedly administer naloxone, which may be difficult given your level of discomfort.
Look into the rapid detox protocol using a form of anesthesia and naltrexone (usually I believe, sometimes naloxone or more often both). What you propose is essentially the same idea, just without the anesthesia - and there is a good reason they want to put the patient under during this process. There are a lot of concerns over the safety (more related to the anesthetic I believe), practicality (cost) and efficacy (you still have to "recover" because all this really seems to accomplish is getting the drugs out of your system a bit faster).
If you feel like trying it, hell go ahead. I would honestly recommend keeping something like buprenorphine or methadone (or your DOC even) on hand if you find you can't handle the naloxone, better yet the whole department of comfort meds (diazepam/clonazepam, gabapentin/pregabalin, clonidine, loperamide, cannabis, valproic acid, etc. etc.). I'd really advise against this though. I don't see it as a matter of testing your resolve when it comes to abstinence - I see this as an issue of concern to your life and health much more so.
If you do anything discuss your plans with someone in your real life, ideally someone who can check in on you if not hang with while you go through hell to help nurse you and be there to get help should any issues arise. Best case scenario, you do this under a medical professional's care, like a registered nurse or GP, but I don't see that happening (cause they'd be liable and I doubt they'd agree unless they're your good friend or something and doing it undercover).
If you decide to go ahead, be very, very careful about any preexisting conditions you might have, particularly seizure history or heart problems. It's not so much an issue with normal w/d generally, but you won't be trying normal withdrawal here. Please be careful.
I don't see the potential problems with you idea for kicking being a matter of gritting your teeth or dealing with the pain. I'm curious, what have your other experiences with withdrawal, detoxing or precipitated withdrawal been?
And here's some random article about rapid detox I found:
http://www.medpagetoday.com/Psychiatry/Addictions/41908
