So, you gotta wonder, is the Erlking PART of the Wild hunt? So, if Harry did call up the hunt, would the EK automagically come with it?
No and Probably. I dont think he, or any actual particular individual, are permanent parts of the Hunt. Rather I think the Hunt is a more of a Primal possessing Spirit (given it's pseudo-sentience, possibly a transient Mantle) that gathers up sympathetic vessels including Thomas and Erl and anyone who takes the 1st or 4th options when it comes. Meanwhile, the Erlking seems willing to show up any time it sounds like a good Hunt. Note how he (a summer-side Wyld fae) still showed to "defend" when the Winter Wellspring was attached; he had no stake or connection, but rather I think he just saw it as an opportunity.