I found this concise summary of the consequence of corneal bulging (keratoconus)
"As the cornea becomes more irregular in shape, it causes progressive nearsightedness and irregular astigmatism to develop, creating additional problems with distorted and blurred vision. Glare and light sensitivity also may occur."
That sounds pretty awful, but I would hope the hybrid contact lenses do a reasonable job on correcting the aberrations, so I would suppose glare and reduced contrast might be the first concers here.
Glare is caused the stray scattering of light within the eye but it's manifestations might be quite variable on where the cause is located. I guess due to the lasik reference it might be corneal scaring, (but I have heard of other complications.) Cutting out light from the side may help. Some binocular eyecups sit well in the eye socket doing a good job but if might help to use horned/winged eyecups for some models. Some makes supply such accessories but Field Research, amongst others supply generic versions.
In normal viewing light will impinge on the whole of the conea and potentially contribute to glare. The exit pupil and apparent field of view will potentially reduce the area of the cornea involved. Using something like a 10x32 with a 3.2mm exit pupil might improve things a bit too, though small exit pupils do reduce low light capability.
Generally in optics, the blue end of the light spectrum is most easily scattered. It can improve contrast to filter out shorter wavelengths. I wouldn't know if it would help with this condition. Do amber or yellow tinted sunglasses improve contrast? If so, then it might help to choose a binocular with reduced blue transmission. High blue transmission cost money for the more complex lens and prism coatings, so this might lead you to try relatively cheap models, including some with silver coated prisms instead of dielectric.
Of course this is all guess work, but putting these things together, it might be something like the Sightron BSII 10x32, fitted with after market winged eyecups might just be a reasonable place to start. The good news is if you find the right deal it might cost you less than $200. I'm sure there are many alternatives that might meet your specific needs, but I would hesitate in recommending the most expensive models on the rack as the first step.
David