Why does hemoglobin electrophoresis in sickle cell trait gives two bands but not three?
This is a really interesting question and again reminds of how ignorant we are about things as subtle as this which otherwise should have been an obvious question.

First of all, although there seems to be no confusion, I'd like to clarify that we are talking of the protein being separated on electrophoresis. Hemoglobin electrophoresis classically refers to the cellulose acetate/citrate agar electrophoresis where the RBC hemolysate is used as the sample. This method uses the fact that, hemoglobinopathies generally have mutated Hb which differ in charge from the original HbA. Charge is a property of separation in cellulose acetate electrophoresis unlike SDS-PAGE where separation is on the basis of mol. wt. 

We can also do a DNA electrophoresis (digressing) to identify a Hbpathy after digesting with restriction enzymes which would give a different results, depending on the restriction enzyme that has been used to differentiate the mutation from the original. Here is an example using MstII for RFLPs. This isn't the topic of discussion here. Our sample is the protein, and not the DNA.

So it is important that the electrophoresis we are going to talk of, is of a protein electrophoresis based on charge.

Considering the sickle cell trait, let us clarify the terminology now. HbA is normal hemoglobin tetramer which composes of $\ \alpha_2\beta_2^A $. HbS on the other hand is $\ \alpha_2\beta_2^S $. The terminology you have used not only isn't standard but also is conveying a different meaning. Now logicaly, in a person with sickle cell trait, one should be expecting a hemoglobin with both different beta chains. And it does happen! Literature calls it the assymetric hybrid. Let us indicate it with AH which is $\ \alpha_2\beta^A\beta^S $. Let's sum this up a bit.

HbA = $\ \alpha_2\beta_2^A $

HbS = $\ \alpha_2\beta_2^S $

AH = $\ \alpha_2\beta^A\beta^S $

The assymetric hybrid that has been mentioned here is only for the sickle cell trait. There are many other hemoglobinopathies and in their heterozygous states one can obtain other assymetric hybrids with a different beta chain.

Now that we know that assymetric hybrids are formed, why do they not show up on the Hb electrophoresis? 

It is generally believed that under physiological conditions, at equilibrium, hemoglobin is a mixture of tetramers ($\ \alpha_2\beta_2$) and dimers ($\ \alpha\beta $) as $$\ \alpha_2\beta_2 ⇌ 2\alpha\beta $$ The dissociation equilibrium constant for oxyhemoglobin is approximately $\ 2x10^{-6} M $ (heme) and is much less for deoxyhemoglobin. The asymmetric hybrid is generally not isolated by conventional techniques, since the tetramer-dimer dissociation constant is much lower in-vitro than in-vivo . Thus the asymmetric hybrid dissociates into dimers $\ \alpha\beta^A:\alpha\beta^S$ and reassociates into more symmetric forms during the separation process.

There are slightly modified methods to isolate the assymetric hybrids under deoxygenated conditions such as isoelectric gel focussing, cross linking the chains, etc. The amount of AH formed in proportion to HbA and HbS can be measured in these kind of experiments and has been found to be comparable to the expected binomial distribution of $\ a^2 + 2ab + b^2 $. This implies that the stability of the AH is comparable to HbA and HbS.

Hope it helped.

To delve further, here are a couple of reference that should be of use.


  
  HF Bunn and M McDonough (1974), Assymetrical Hemoglobin Hybrids.
  An Approach to the Study of Subunit Interactions, Biochemistry,
  13(5)989-93.
  MK. McCormack, TG. Westbrook, RM. Paull &amp; S Berman (1981),
  Identification of the Asymmetric Hybrid of Human Oxy Hemoglobins A
  and S (α2βAβS) at 4° C by Cellulose Acetate Electrophoresis,
  Hemoglobin, 5(3)251-56.