Pyruvate dehydrogenase: Apparently anomalous NAD/FAD redox reaction
Summary
The redox potential widely quoted in text books for the half-reaction reduction of FAD to FADH2 is indeed inconsistent with the observed direction of several reactions involving this coenzyme, including its reduction of NAD+ in the dihydrolipoamide dehydrogenase reaction. The reason for this is that the value quoted is determined for free FAD in aqueous solution, which does not equate to the situation within the enzyme. In its tightly bound state in flavoproteins, the immediate environment (and, in some cases, covalent linkage) alters the affinity of the flavin ring for electrons, hence causing the redox potential to deviate from that in solution. The extent and direcction of this deviation differs in different flavoproteins.
Restating the problem
The basic idea of chemical theromodynamics is that reactions are energetic favourable if the (Gibbs) free energy change is negative (see e.g. Section 14.1.1 of Berg et al.). Although the actual free energy change, ΔG, is influenced by the relative concentration of the reactants, the usual starting point for assessing this in biochemistry is the ‘standard’ free energy change at pH7 and 25˚C — ΔG0′. In the case of oxidation/reduction reactions the standard oxidation–reduction potentials E0′ of each of the two half reactions are usually known (as they can be measured in solution), and one calculates the ΔG0′ from the oxidation–reduction potentials of the two half reactions, using the formula:

ΔG0′ = –nFΔE0′

What is important for this discussion is not so much calculation of actual values, but determining their sign, as for a negative ΔG (reaction energetically favourable) a positive ΔE is required.
Let us consider an example using the much abbreviated version of Table 18.1 of Berg et al. Biochemistry, below:
Oxidant     Reductant       n     E´o (V)
Succinate   α-Ketoglutarate 2   – 0.67
NAD+        NADH            2   – 0.32
FAD         FADH2           2   – 0.22
Fumarate    Succinate       2   – 0.03

Consider one of the energy-generating oxidation reactions of the Krebs tricarboxylic acid cycle:
α-ketoglutarate + NAD+ → succinate + CO2 + NADH
This is made up of two half reactions:
(+ 0.67 V) α-ketoglutarate  → succinate
(– 0.32 V) NAD+ → NADH
which gives an overall ΔE0′ of  + 0.35 V, i.e. a negative ΔG0′, which is consistent with the actual direction of the reaction and the production of energy. (Actual worked numeric examples can be found in Berg et al.)
Now if we consider the dihydrolipoamide dehydrogenase reaction
NAD+ + FADH2 → NADH + FAD
The two half reactions are:
(+ 0.22 V) FADH2 → FAD
(– 0.32 V) NAD+ → NADH
This gives an overall ΔE0′ of – 0.1V, i.e. a positive ΔG0′, which, as the poster astutely points out, is not consistent with the direction of the reaction.
Resolving the apparent anomaly
Berg et al. do, actually, remark on this in section 17.1.1 on the pyruvate dehydrogenase complex:

This electron transfer to FAD is unusual, because the common role for
FAD is to receive electrons from NADH. The electron transfer potential
of FAD is altered by its association with the enzyme and enables it to
transfer electrons to NAD+.

So, how is “the electron transfer potential of FAD altered by its association with the enzyme”? The quotation from Ghia and Massey1 provides a general summary:

The pyrimidine nucleus of the three-membered ring system is electron-deficient and can be viewed as an ‘electron sink’. In thermodynamic terms, any interaction which tends to lower its electron or negative charge density will increase the redox potential. The reduced flavin consists roughly of an electron-rich phenylenediamine moiety fused with a (4,5-diamino)-uracil. The latter is the moiety in which the negative charge (i.e. two electrons taken up by the oxidized flavin) is localized and, most importantly, stabilized. The degree to which this negative charge is stabilized or destabilized is an important factor governing the redox potential. Thus while a positive charge in the protein around the pyrimidine ring will contribute to increase the redox potential, the presence of negative charge or of a hydrophobic environment will lower it.


In the specific case under consideration — dihydrolipoamide dehydrogenase — it has been established2 that a positively charged lysine-57 sidechain (numbering for the A. Vinelandi enzyme) that increases the redox potential (with, perhaps, a small contribution from the backbone N of alanine-326):

[FAD environment in dihydrolipoamide dehydrogenase from A. Vinelandi. NAD is shown in light blue, except for the O and N atoms at the end of the flavin ring interacting with the protein. A section of the latter is shown as a wireframe in salmon pink without sidechains, except for lys57 and ala326. (Own diagram from 3LAD.pdb)]
A second apparent anomaly and its resolution
In drawing attention to the apparent anomaly in the reduction of NAD+ by FADH2, the poster suggested that this would imply one should by able to obtain enough energy to produce 3ATPs from reducing FADH2. The problem with this is that one actually needs FAD to have a lower redox potential to perform the reduction of fumarate that occurs in the Krebs cycle:

This can be seen by considering the redox potentials in the table above:
succinate + FAD + → fumarate + FADH2
The two half reactions are 
(–0.22 V) FAD → FADH2
(+0.03 V) succinate → fumarate
Thus, there is an overall ΔE0′ of – 0.19V, i.e. a positive ΔG0′ — another anomaly!
Berg et al. are rather coy on this one. They do not comment, but in their Table 17.2 of free energy changes in the steps of the TCA cycle — where all the other values are given as exact figures — they quote  “~0” for this step.
This implies a higher (more positive) redox potential for FAD than the –0.22 quoted in the table — certainly not the lower one of dihydrolipoamide dehydrogenase.
I turns out that this is, in fact, the case. In succinate dehydrogenase (aka as fumarate reductase) FAD is covalently attached to a histidine residue (his44 in E. coli) at the other side (position 8) of the flavin ring3.

[FAD environment in succinate dehydrogenase from E. coli. NAD is represented as previously. A section of the latter is shown as a wireframe in salmon pink without sidechains, except for his44 and a backbone N. (Own diagram from 1L0V.pdb)]
An extensive review4 of the redox potentials of covalently and non-covalently bound flavoproteins shows that covalent attachment increases the redox potential.
Final observations

Chemistry is a powerful tool for understanding biology at the
molecular level. Biology is subject to the laws of chemistry, but the
chemistry of the special environments encountered (protein interiors,
spaces compartmented by membranes) need taking into consideration,
something that the classically trained chemist, new to the subject,
may not fully realized.
Elementary textbooks often have to walk a fine line between
comprehensibility and truth. The student of biological chemistry
should realize that the initial simplifications that enabled him to
get an overall grasp may mask greater complexities. One of the
intellectual satisfactions in science can be found by resolving these
complexities in a rationale manner.

References
1Ghisla and Massey (1989) Mechanisms of flavoprotein-catalysed reactions
2Maeda-Yorita et al. Biochemistry (1994) 33, 6213–6220
3Kim and Winge (2013) BBA 1827, 627–636
4Heuts et al. (2009) FEBS Journal 276, 3405–3427