Q:

Conditional deniability of a natural asteroid belt with an atmosphere?

Is it unreasonable to suspend one's disbelief that an asteroid belt naturally formed with enough mass distributed through the belt to justify the formation of a belt-wide atmosphere given the following conditions and limitations?

The atmosphere should be Earth-like @ sea level (14.7psi) along the surface of a continuous imaginary tube weaving through the asteroids and around the belt that represents the high-G point of gravity (1G) at any point along the circumference of the belt.  I'm not imagining a perfect, flat toroid.  I do expect the toroid to change shape as asteroids shift around.

NOTE: That imaginary/mathematical tube/toroid is important.  Just as gravity is at its maximum when standing on the surface of a planet sphere and decreases as you move away from that surface in either direction, eventually tapering to zero in outer space or at the center of the planet, the gravity at the surface of this tube or toroid (imaginary, it's the mathematical high-G point) tapers to zero at the center of the imaginary tube or beyond the boundaries of the belt itself.  Yes, this would suggest that the asteroids tend to be dense within the boundaries of this imaginary tube.

The atmosphere is expected to thin as one proceeds from the imaginary torroid mentioned above toward the boundary of the belt.  (If you're travelling from the jelly in a jelly donut toward the powdered sugar, the atmosphere gets thinner, which would make for a very disappointing jelly donut, if you think about it.)  It may thin quickly, that's OK.  I recognize the atmosphere from the shell of my imaginary tube toward the center becomes thin and chaotic.  Let's ignore that.
The belt is more-or-less 1AU from a Sol-type star.
I do not expect any asteroid to have a surface gravity of 1G.  I'm only looking for the aggragate gravity along that previously mentioned imaginary torroid to be great enough to hold 14.7psi gravity in place.  As mentioned, the high-G "surface" (the location in space that surface represents) of the imaginary tube is 1G.
I recognize that Van Allen radiation belts are necessary to keep the solar wind from blowing the atmosphere off into deep space.  Please ignore that requirement at this time.
I also recognize that, without a replenishing source, the atmosphere would escape to space (there being much, much more surface area, so to speak, for this to happen when compared to a planetary sphere).  Please ignore this, too.

The really important part
I appreciate that "suspend one's disbelief" is very subjective.  Where one is willing to believe outright for the enjoyment of the story another is prepared to ceremonially burn the story after dousing it in the blood of three diseased chickens due to the obvious and outright scientific fraud it's perpetuating.
I consider the two limitations above (missing radiation belts and high atmospheric escape) to be the two biggest reasons this wouldn't be factual.  I'm asking if that's enough, or are there other reasons that beggar belief so much that the idea would give readers cancer.
Therefore, the best answer is the one describing the most non-trivial reasons why this wouldn't work.  
From my perspective, if I win the tumbleweed badge for this question (which has never been awarded for this site, not to put too fine a point on it), then I have a winning idea on my hands.

A:

To stay in orbit the asteroids and the atmosphere would have to be orbiting around the sun. 
The asteroids and the atmosphere closer to the sun would have to be moving faster. 
So different parts of the atmosphere would be moving at different speeds.
When two bodies of gas are moving relative to one another there is some friction between them. And any kind of friction will result in kinetic energy of the gas being transformed to heat energy. The loss of kinetic energy will cause the atmosphere to move closer to the sun.
The consequence of this is the at the atmosphere will move to a lower orbit.  And lower, and lower, dragging the asteroids with it. As it falls lower it will heat up. Some asteroids will collide with each other, releasing more energy and heating up the atmosphere further.
You have just invented an accretion disc. If the process is not checked by the formation of a planet in the accretion disk, the eventually the atmosphere will fall into the sun.

