No, I wouldn't do it for a number of reasons.
First, you should not find yourself OOA. There are some some remote possibilities of equipment failure, but for the most part, this is due almost entirely to diver inattention.
Second, you should not be diving without a buddy unless you have a redundant air source. If you are both OOA and OOB (Out of Buddy), you really need to work on your and your buddy's attention to safety.
Third, in most any OOA situation, there is still air in your tank that is not available due to ambient pressures. As you ascend, the pressure in the tank becomes greater than the ambient pressure and air will become available to you. Keeping your regulator in your mouth and your airway OPEN will facilitate you being able to breath off of this increasingly available air supply. Removing your reg to try to breathe off of your BC will flood the reg and require you to purge it before you try to breathe.
Fourth, when you are OOA, you don't need to mess with your buoyancy! You will find that when you are truly OOA, that you will want to exceed that Max 30fpm ascent rate you learned in class. The air in your BC expands as you ascend actually helping you to ascend. Is this dangerous? Not near as dangerous as your imminent drowning. Kick like hell and tear for that surface.
Finally, if you are properly weighted, you have very, very little air in your BC. I often dive with no exposure suit and with little to no air in my BC. There is NOTHING for me to breathe in there! If I just ran out of air, then I don't have anything to put in to start that cycle and I am wasting TIME rather than seeking that big alternate air source you call the surface!
In the beginning days of diving I dove with a J-valve and no SPG. When I eventually ran out of air, I normally pulled the rod which gave me more than enough air to ascend. More often than I liked, I would find the rod was already down. Since I had no concept of diving with a buddy at the time, the surface was my only hope. There are a few of those events still etched clearly in my mind, but the outcome was always the same: breathing that sweet, sweet air on the surface.However, since I started using an SPG, I have to brag that I have never, ever run out of air. Not once. Start your own tradition of not running out of air from this point forward!
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Thank you for this very well put explanation.
The only time I was in real trouble and have a bit of PTSD which I have worked through, was on a dungeness crab dive. I just want to warn people about this.
My weight is perfect (I'm not talking about my belly

But the point is, that there are times that your buoyancy may change and so may the surface conditions be much different than when you went down. Fish are more neutrally buoyant than shellfish!
And "No, I would breath out of a BCD" for all the reasons that NetDoc so wisely and concisely said.
Thank you again NetDoc.