For the BJT in the reverse active region, why shouldn't you dope the emitter and collector almost equally, even though it will increase beta_r? Also, what are some ways you can increase beta_r (in reverse active) or beta_f (in forward active)?
To increase beta_f, the base region is made ultra thin, and the doping is very light compared with the emitter. This yields high injection efficiency. Supergain devices, aka superbeta devices, employ this technique to achieve beta values around 5,000. The tradeoff is that the Vce breakdown is just a few volts, and the collector leakage current is very high. Op amp inputs use these devices.
The beta_f value will drop when the device is built to handle higher voltages. To get high Vce capability, the base region doping is increased, as well as the region being thicker. This reduces beta_f, but reduces the c-b reverse leakage current. Also, heavier base doping decreases rbb', which is desirable.
To get higher beta_r would require light base doping, and/or heavier collector doping. Heavy collector doping would reduce Vbe in the forward mode. In the reverse mode, the emitter acts as the collector, and Vbe is only around 6 - 7 volts, so that "Vce" in reverse mode is 6 to 7 volts.
I'm just curious as to why you want higher beta_r. The bjt is seldom operated in reverse mode. The only application I can recall is TTL. The input to a TTL gate getting yanked low puts the bjt in reverse mode operation IIRC. But beta_r being low is not a problem since the input is the collector (acting as an emitter in reverse mode), and the input signal source provides collector current, not base current. Base current in reverse mode is greater than collector current, since beta_r << 1.