VA hospitals vary greatly in size from less than 40 beds to over 500. The larger the size, the more likely they have a dedicated CL position. Almost none are going to require a CL fellowship. There are very, very few CL fellowship trained psychiatrists in the country and VA's could not fill their positions if they required that. The VA also can't offer more pay for subspecialty fellowship training. Experience in something similar to CL, maybe a dual board in FM, job experience, or even a fair amount of residency experience would be more than sufficient.
In my experience, most VA's have dedicated inpatient psychiatrists who also do CL since they both don't require dedicated scheduling unlike outpatient. Outpatient providers have certain scheduling metrics that don't mesh well with inpatient work where they don't apply. The whole see your inpatients in the morning and outpatients in the afternoon is relatively rare in the VA, although I'm sure it exists. You seemed to describe something like outpatient CL. In the VA, this is called Primary Care Mental Health Integration where a psychiatrist is stationed in the primary care clinic and available all day to assist and assess with mild to moderate mental health patients presenting to their PCPs. This also has the possibility for CL med floor consults since it also isn't subject exactly the same to outpatient scheduling metrics, but again, I see inpatient people doing CL a lot more often. It just works better to mix it with inpatient in terms of flow because many (most?) of the med floor consults will be at least indirectly pushing to transfer behaviorally problematic med floor patients to the psych unit. In terms of specialty care, I'm not entirely sure what you mean. The VA is mostly a generalist organization, excepting things like PTSD or those conditions that are extremely common amongst veterans. It contracts out the rare stuff. There are VAs that do things like organ transplants and bariatric surgery, but these are quite rare.
For more concrete advice, you can always check usajobs.gov and contact the physician recruiter for the site listed on the job even if it isn't quite the exact job you were looking for. Tell them what you are looking for and please remember that everywhere is hard up for in person providers. So whenever you're reaching out to a physician recruiter, please immediately mention that you plan to be in person. It will get you higher up on a lot of lists. A lot of applicants seem to assume that all positions are at least potentially virtual and there's a lot of pressure from directors to avoid hiring/making new virtual positions whenever possible since most of our veterans are still not digital natives.