Sorry that I missed this thread. I'm late to the dance again.
OK, there are no known surviving bowie, 'hawk, and bowie/hawk lineages. From what we know of the time period, extant "systems" tended to be very basic. While there is some degree of social exclusion (don't show the outsiders) that doesn't seem to have been the primary reason for the die-off of what skills there were. Instead, the reason seems to have a lot more to do with simple facts of technological advancements making those weapons (and therefore their skills) obsolete. Most of the social groups in question ("native" Americans, Colonials, Settlers, Military, Naval) tended to be rather practical and pragmatic. Their lives were already full of just surviving, making ends-meet, and trying to raise the next generation that they frequently had to be very choosy about what skills were considered important enough social/heirloom artifacts to be passed on to the next generation. The death of Bowie/Longknife & 'Hawk traditions occurred for much the same reasons that caused the death of most other fencing skills.
This is made worse because of other social stigmas at the time. While Fencing in general was considered gentlemanly, there was a huge social stigma in the U.S. against the knife, particularly following the "Bowie Knife Craze" of the mid-19th Century (where news media of the time reported any fixed bladed knife over 6" as a "Bowie Knife"). While, for a period, there were some schools, notably in New Orleans, teaching Bowie, there are no known period Bowie Knife manuals. The same goes for the Tomahawk.
Having said that, there are some notable sources of secondary evidence for how the 'Hawk was used, mostly from period accounts and the occasional medical record. Using these sources, a good understanding of what other skills were available and in practice at the time (such as well known and understood period Fencing and other native fight traditions), and a healthy dose of "experimental archeology," it is possible to develop a set of skills that are
probably pretty close to what was practiced in period.
This is the direction that the afore mentioned Dwight McLemore has used in his work on the Bowie Knife, Tomahawk, and their dual use. And, yes, the results are very good. It has produced a system which is unified in theory and method, pressure tests well, and survives third party scrutiny (in no small part because he is very up-front about his sources and methods).
For full disclosure, I have studied with Dwight and several of his students and consider them to be friends. I have Dwight's material, both books & video, and I teach his method at my Club.
Peace favor your sword,
Kirk