Is the airspeed REALLY increasing in the bumps,
or not?

1) I doubt it's a pitot tube issue. Studies have shown that Pitot
tubes are very accurate until beyond the stall AOA.

2) A reduction in static pressure would produce the same phenomenon
and occurs more easily than a Pitot tube problem.

3) Possibly a function of the static longitudinal stability of an
aircraft. In response to an increase in AOA, the natural stability is
to pitch down.

4) While it's true that there is horizontal windshear, the literature
on aircraft structures assume it's minor compared to vertical
windshear. Still might be enough to affect the Pitot tube, but on
average you'd think it'd average to zero.

5) I'm no weather expert, but the air does accelerate and decelerate
as it moves to and from pressure systems. Seems reasonable that
changes in velocity would generate turbulence, rather than vice versa.
Perhaps a different weather system would generate different results.