jantze, someone else had asked me essentially the same question, so I'll answer here.
As far as I could tell, the TPA Ivy is an I/V converter for the Opus DAC (converts the DAC's balanced current outputs to balanced and unbalanced voltage outputs). It is not a general-purpose balanced/unbalanced conversion device.
If you build a balanced β22, then you could just use the Ivy's balanced outputs to feed the β22. There is no need to use the Ivy's unbalanced outputs.
You may want to use a source that only has unbalanced outputs (e.g., a CD player, tape deck or whatever) with the balanced β22. This is where an unbalanced/balanced switch comes into play. But since the balanced β22 will accept only balanced inputs, before you could switch anything, you need to convert the unbalanced source' signal to balanced. This is because the inverted "cold" signal for each channel is missing. An unbalanced-to-balanced line driver circuit (which the Ivy isn't), or a special transformer made for this purpose must be used in order to derive that signal. That will then allow the balanced headphones to remain "balanced" (i.e., hot and cold signals of oposite phase at each end of its drivers).
An example of an active unbalanced-to-balanced line driver is a circuit built around the DRV134 chip. Lundahl, Jensen, Cinemax and others make passive conversion transformers, but they are very pricey. Both methods have their benefits and drawbacks.
If you don't have all balanced sources, then you should consider building the 3-channel active ground β22 instead. It gives you virtually all the benefits of balanced (except the doubled voltage swing), and you don't have to deal with unbalanced-to-balanced converters which would degrade the signal.
As for amplifier gain, I am not familiar enough with the D2000 or D5000 to make a comment about it.
