There's really a lot to unpack here. (I rewrote this response thrice...) To spoil the answers straight away, and perhaps to address some misconceptions upfront:
> If a brown dwarf is right on the edge of stardom would it start protium fusion in only the part of itself that is the right pressure
The nuclear cross-section of protium fusion in astronomical bodies is determined by temperature, pressure, and density. These variables are in turn dictated by the total mass of the object in question. Brown dwarfs never have sufficient mass for protium fusion, so they never undergo any protium fusion whatsoever. This is a hard-and-fast boundary for stardom.
> or would that ignition precipitate a pressure wave through the entire body forcing fusion to begin everywhere
The three variables above are not uniformly distributed throughout the body; there is a maximum at the centre, and fusion only happens here. Even in the Sun, about 99% of the fusion happens within about a quarter of the radius from the centre. The temperature drops rapidly thereafter, reducing the nuclear cross-section of the proton-proton chain to essentially zero. At the photosphere (surface), the temperature is ~5777 K, which is a decidedly Earthly temperature (lightning bolts are ~ 30000 K). So no, fusion does not happen everywhere.
Now, we need to discuss star formation and why brown dwarfs have never experienced protium fusion at any time in their lives. Star formation is still a very active area of research, debate, and fitting models to empirical study, and this is especially true for the detailed interior and structure of protostars and pre-main-sequence stars; hence, this is going to be quite surface-level (pun not intended).
Collapsing molecular clouds form stars. The total mass of a given cloud (or a particular region of it) sets an upper bound on the resultant object, because the total mass dictates the gravitational potential energy and hence the terminal velocity of the matter, and hence the rate of matter infall at the centre before the cloud dissipates. If the mass is low enough, the central object will become a brown dwarf, or even a large gas giant and a 'rogue planet'.
As the gas cloud collapses, the central region increases in density, temperature, and pressure, but no fusion occurs yet. Conservation of angular momentum forms a circumstellar disc, and material continues to fall onto the central region. As long as this infall continues, the central region is called a protostar. At some point the mass of protostar crosses the boundary needed for deuterium fusion; if the infall stops here, the result is a brown dwarf. If this infall continues, the mass increases beyond the boundary (~80 Jupiter masses) needed for protium fusion, and protium fusion can begin. When the infall stops and the circumstellar disc largely dissipates, the result is a pre-main-sequence star.
Note that both these very young pre-stellar objects are not yet at hydrostatic equilibrium, and are still comparatively rarefied (or 'puffy') compared to main-sequence stars; they are still collapsing, and the temperature, density, and pressure at their cores continues to increase. Only when this equilibrium is achieved and gravitational collapse is halted do stars begin life on the main sequence.
Now, it should be evident why brown dwarfs never experience protium fusion: at no point in their lives have they ever had any region in their interior hot, dense, or hyperbaric enough to have a high enough nuclear cross-section for protium fusion. At their formations, they were simply not massive enough; they continue to collapse, which admittedly provides a considerable power output—surface temperatures are ~1000 K. The largest brown dwarfs experience deuterium/tritium/lithium fusion into helium, but this also stops over time.
(Side note: in my opinion the word 'brown dwarf' is a bit of a misnomer, because look at how bright molten iron (~ 1500 K) is even in broad daylight[1]; now imagine an object ten to twenty times the radius of Earth, emitting this much heat from every square millimetre. If you approached a 'new' brown dwarf it would cast a lot of light.)
[1]: https://commons.wikimedia.org/wiki/File:Scunthorpe_Molten_St...
