You will need to make sure that your kit is set up correctly to use the small caps variant. The only way to determine which font is loaded in the hoefler child stack is via the weight. If you can make sure that the small caps variant is the only one in your kit at that particular weight and put it in its own font family child stack then you should be good. Steer clear of the Font Styles small caps setting though as that does not interact with the hoefler kit settings but merely applies a CSS synthetic small caps as described above.
The only problem I see you running into is if you need Archer at the same weight in non small caps (i.e. also have that in your kit) then the browser will probably just default to loading the normal (book) variant.
I’ll have a look in my h&Co dashboard as soon as I get a chance but I’m not sure if any of my 5 licensed fonts have small caps off the top of my head. If they have, I will experiment and get back to you.