After many failures, I finally succeeded in making a functional helium discharge tube. The main problem throughout the development process was purifying the helium in the tube. The helium from the balloon kit is only 80% pure, so the remaining 20% of air needs to be removed somehow. I initially tried to replicate the process I used for the glow tube, where I ran excessive current through the tube to overheat the titanium electrodes and use them as getters. I couldn't achieve this without cracking the glass or ruining the feedthrough seals though, and indeed the original glow tube no longer functions. I eventually made a hairpin electrode for each end out of 0.8mm titanium wire, where one end could be operated as a glow tube (to get a rough estimate of the pressure inside of the tube) and the other end had a 0.3mm titanium filament that could be heated orange-hot to act as a getter once the tube has been sealed off. A diagram of this can be seen below.
The finished tube is 8 inches long and 6mm in diameter. I used some of the old flint glass tubing I had ordered previously from Teacher Supply Source for the body of the tube, as I discovered it sealed well to the 5mm neon sign tubing which I used for the evacuation stem. The pressure inside the tube is unknown, but based on the characteristics of the glow tube end, I estimate it to be between 1 and 5 torr. The tube operates on 2.5KV AC with a 10nF capacitor in series to limit the current to approximately 10mA. It glows pink initially, but after running for a few seconds it gets slightly brighter and becomes mostly orange-white, with a slight pinkish hue. It is pleasant to look at, however it does begin to flicker eventually. I believe this is in some way due to the geometry of the electrodes, as the flickering coincides with the glow jumping to any bends or sharp points on the wire.
In operation, the tube glows bright enough to be used as a (somewhat dim) desk lamp, and the majority of the tube stays fairly cool. The glass near the electrodes gets very hot however, and some sputtering of the electrodes can be observed (visible in the second picture above). An improved design might use larger diameter tubing around the electrodes, to reduce the chance of the glass cracking from thermal shock. The sputtering also leads me to believe that the filament getter may not be necessary, and that simply running the tube could absorb any air in the tube over time.