Another possibility would be to not screen them, provide them with a high level of free care as a reward for their service, and just make the system work efficiently. Whether the government is footing the bill, or an individual is, should make no difference at all to the timeliness or level of care they receive. If the problem is paperwork, that should be targeted first.
There is a lot of medical care available (I think "stay on Tricare", etc.) with less hassle. Most of the disability claims are for ongoing cash payments to replace job income, which is more at risk of fraud (and in some cases genuinely uncertain and requiring investigation).
