If I were to take that shot I would center punch right through the spine into the vitals. IMO attempting to shoot to the side of the spine only increases your chances of missing the vitals. I personally would not hesitate to take such a shot if using a fixed blade broadhead, especially if it has a chisel point. I would be reluctant with mechanicals. I don't intend to turn this into another mechanical vs.fixed debate, but in regards to this shot, the only problem I've ever had with a mechanical is breaking off or bending when hitting against bone. IMO shooting through the spine invites mechanical broadhead failure.
Perhaps more importantly, and the most common mistake made on such a shot is the proper aim spot in order to hit the vitals. Too many aim directly between the front legs as if they were looking at where the vitals lie from a side shot, or shooting in a direct line straight down. If you aim there you will likely hit to far forward missing the vitals. I will try to illustrate with some crude MS paint.
You have to take in account the depth of the vitals in the chest cavity, and the arrow flight angle needed to hit that location. I have provided both the side angle and an aerial view showing the relation between the angle needed and the aim point on the body to hit the vitals at such an angle.
From your 3 yard scenario you probably need to aim slightly behind the vitals in order to hit them.
As the deer walks away from you, you will need to aim farther back on the body to maintain an angle that will connect with the vitals.
I'm sure there will be differencing opinions as to whether you "should" take that shot. Ultimately that's up to you. I hope this helps, should you decide to do so.
Live to Hunt, Hunt to Live.