Ah, I see! Given that, I was able to solve my problem by defining MessyAction(_) as a CONSTANT in the SystemMC spec, then in the model I define the value of the MessyAction(_) constant as MessyActionImpl(_). See below, differences highlighted:
------------------------------- MODULE System -------------------------------
EXTENDS
Naturals
CONSTANTS
MessyAction(_)
VARIABLES
Counter
-----------------------------------------------------------------------------
TypeInvariant ==
/\ Counter \in Nat
Init ==
/\ Counter = 0
Increment ==
/\ Counter' = Counter + 1
/\ MessyAction(Counter)
Next ==
\/ Increment
=============================================================================
------------------------------ MODULE SystemMC ------------------------------
EXTENDS
Naturals
CONSTANTS
MessyAction(_),
MaxCounterValue
VARIABLES
Counter,
PastValues
ASSUME MaxCounterValue \in Nat
-----------------------------------------------------------------------------
MessyActionImpl(c) ==
/\ PastValues' = PastValues \cup {c}
S == INSTANCE System
TypeInvariant ==
/\ PastValues \subseteq Nat
/\ S!TypeInvariant
Init ==
/\ PastValues = {}
/\ S!Init
Increment ==
/\ Counter < MaxCounterValue
/\ S!Increment
Next ==
\/ Increment
=============================================================================