I still think that a lot of the resultant sound you get is in how the bass drum really sounds in real life. Also when you say it sounds punchy etc it might sound like that up higher where the drummer is but get your head right down in front of the hole and have a listen there, it is often a different story.
It is hard but you can kneel down there and operate the beater by hand while listening close to where the mic is. Both heads effect the sound a lot. Also the damping inside also has a big impact. It should be touching the edges of the heads to a certain extent but not kill the sound either. I often put a small brick on the damping to keep it in place.
The front head can still effect the sound too. The batter head obviously has a big impact on the sound. Tension etc, pitch etc. What type of head is the batter head? There are some clever batter bass drum heads that have an extra ring around the edge to control unwanted rings etc..
I am lucky because I play a Sonor kit and really those drums are better than nearly everything else on the planet. I say that because even if my bass drum is poorly tuned it still sounds killer. That must say something.
(I can put an SM58 in there and still get a killer sound) If I spend time tuning it sounds ridiculous! So get the drum sounding as best as you possibly can after all if that is not right nothing will help you from that point on. You should be able to get a killer recorded sound with no processing virtually. It can be done.
As far as the mic is concerned I am not a great fan of the AKG D112 either. Sure it is a good mic but I find a Shure B52 sounds quite different and better. Not sure about the Heil mic. We have an Audix D6 at work where I teach and I have used that and it is also great. I also have the Antares Mic Modeler plugin and I find that it is quite excellent at changing the sound from the recorded mic in pretty much anything you want to hear! That plug works!!!
Rimshot mentions below to put the mic in front of the hole. Try that for sure. I put the mic right inside the hole inside the bass drum. That is where I get the better sound for me anyway. You don't need to isoloate the bass drum with any covers. But saying that I have also got great results buy putting a large piece of heavy carpet right around the bass drum to form a cannon like extension in front of it and put the mic outside the hole but still inside the tunnel obviously. If you do that you need to block the end of the tunnel somehow as well to keep the isolation.