After a little investigation, this is already done.

The mobileview API can accept prop=pageprops parameter, which'll make it mix in the request page properties to the response. Adding pageprops=disambiguation to the query parameters does indeed expose the property just as the Disambiguator API does.

Compare:

https://en.wikipedia.org/wiki/Special:ApiSandbox#action=mobileview&format=json&page=Barack_(disambiguation)&prop=pageprops&pageprops=disambiguation

{
    "mobileview": {
        "pageprops": {
            "disambiguation": ""
        },
        "sections": []
    }
}

with https://en.wikipedia.org/wiki/Special:ApiSandbox#action=mobileview&format=json&page=Barack_Obama&prop=pageprops&pageprops=disambiguation

{
    "mobileview": {
        "pageprops": {},
        "sections": []
    }
}

Now, if the Disambiguator extension isn't installed, then the disambiguation page property won't exist. Fortunately, the mobileview API seems to silently ignore undefined page properties, e.g. https://en.wikipedia.org/wiki/Special:ApiSandbox#action=mobileview&format=json&page=Barack_Obama&prop=pageprops&pageprops=foo