Best way to "bend" an object

First, let me point out that the figure you are trying to draw--since the shanks of the arrows are just uniform in thickness anyway--can be done by simply applying the Arrowheads Effect to a circular arc:

Also, I did try creating a custom brush and it works in principle but it makes some changes and distortions to the proportions and angles of my custom object that aren't acceptable.

Sounds like you're looking to bend your group without distorting it's contents.  There's not really any way to do that.

Yes, there is. In this kind of problem, remember two things:

A single Path can have multiple strokes applied. Those strokes can be ordinary Strokes, Brushes, or combinations of those.

The position and spacing of a tile in a Pattern Brush is controlled by an invisible noFill/noStroke rectanguar path at the bottom of the tile's object stack. You can therefore prevent end tiles of a Pattern Brush from being distorted by altering the width of that invisible rectangle such that most of the artwork is outside of it:

Again, the arrows of your figure have ordinary straight shanks. So an ordinary Stroke of a heavy weight serves to draw them. But what if you want the shanks of the arrows to be non-uniform?

Remember that you can apply multiple Strokes to a single path. That means you can apply multiple Brushes (and therefore multiple Brush types) to a single path. Study the white arrows (item 6) in the illustration below. They have tapered shanks, yet they have uniformly-sized and shaped heads and elliptical tails. The illustration shows exactly how they are built. They are a combination of an Artbrush (for the shanks) and a PatternBrush (for the heads and tails) applied to a single path. Saved as a Style, I can use it to draw a "lasso arrow" of any shape/length/weight with the desired result:

JET

SD,

Maybe a few images showing the originals and the bent shapes may help the helpers suggest a way to overcome the distortions of the latter.

Or, especially if you wish to be able to work with each object individually, or to adjust the lengths, you may just make each into an Artbrush (drag and drop upon the New Brush button in the Brushes palette), and apply to each part of a  circle cut in three.

My goal is to take 3 of my custom rectangular objects and bend each one along an arc that represents roughly 1/3 of a circle (~120 degress each) and position them so they form a 3 phase, circular cycle graphic.

Should it be precisely 120 degrees or not? The best way of cutting depends on it.

You can try using effect->warp, but that might mangle too much. [edit - also might be hard to apply to a precise circle]

You can try making a custom brush out of the 3 of them laid end to end and apply that to a circular path.  Lay them out in a line and drag them onto the brush panel.  Choose Art Brush from the dialog that comes up.