To answer this I used a Zemax model of the medium resolution mode of EXES (echelle chamber only) where the echelle grating is turned into a flat mirror and a 'pupil imaging' KRS5 lens of focal length 38mm (at 633nm wavelength) is placed between the last turning mirror and the detector.
This layout is sketched below:
I placed an 88mm object 2760mm away from the entrance lens to simulate the CHP and I used wavelengths of 633nm and 13.7microns.
Imaging at the Detector
The quick answer to the initial question is that EXES will image the CHP if we move the pupil imaging lens away from the detector: perhaps about 3mm for 633nm wavelength and about 12mm for 13.7microns wavelength. The figure below shows the 88mm object imaged at 633nm wavelength into a roughly 2mm image while the 13.7micron wavelength is defocused. (This all depends on where the lens really is and how you measure the distance from the lens to the detector. I measured from the center of a 5mm thick lens to the face of the detector.)
In the image below I show the effects of moving the pupil imaging lens away from the detector in two 3mm steps. I've indicated the changing distance from the detector to the outboard vertex of the biconvex lens. I've kept the total distance from the turning mirror to the detector constant at 191.23mm - which may or may not be accurate.
New Pupils and New Foci
I defined the entrance lens as the aperture stop, which created pupils at the turning mirror before the slit and also about 12mm before the pupil imaging lens.
At 633nm wavelength foci within the instrument now occur about 112mm beyond the cold stop, also about 1/3 of the way from the grating toward the second hit on the paraboloid and on the detector.
At 13.7microns wavelength foci occur about 280mm beyond the cold stop, about 100mm beyond the second hit on the paraboloid and on the detector.
Caveats
Although this system will image the CHP, it is extremely inefficient.
Probably the biggest loss is due to small entrance lens and long distance to the CHP, but, in addition to that, the beam becomes quasi-collimated at the slit (about 14mm in diameter) which will cause losses if any slit is in place (see image to right.)
In addition, as always, the model is not the instrument and distances in the model may not always correspond to their counterparts in reality.
