Most NP-completeness proofs I have seen pertain to proving that a problem on a class of instances is NP-complete. E.g., satisfiability is NP-complete on the class of instances with clauses having three literals. Usually these proofs exploit the room that the definition of the class allows to reduce every instance from a known NP-complete class to an instance in the class of interest.
The question I am interested in is somewhat nonstandard: suppose I have a sequence of instances $P_1,P_2,...$ with size of input $n_1,n_2,...$. The size $n_k \rightarrow \infty$ as $k\rightarrow \infty$. I want to claim that this specific sequence of instances is NP-complete. I know the instances are in NP, only NP-hardness remains to be shown.
Here's the challenge. Because I have a specific instance for each $k$, I don't have adequate room in the definition to reduce the known broad classes to this class. Furthermore, the sizes of my instances are also fixed, which restricts me further. Conceptually, I think I will have to argue that for every instance in some NP-complete class, there exists a $k$ such that this instance reduces to $P_k$. But I am not aware of techniques that reduce known instances to instances of interest with such specificity.
So my question is: are there techniques for proving NP-completeness/NP-hardness type results for a specific sequence of instances?
In case it helps, the instances $P_k$ are 0-1 integer programming problems with specific constants; in fact the constraint matrix of $P_k$ contains that of $P_{k-1}$ as a submatrix for all $k$ and the number of variables $n_k = 2^k$. Notice also that the set of instances $P_k$ has cardinality of $\mathbb{N}$, which is also the cardinality of the number of SAT instances (or finite graphs, or integer programs etc); thus as far as I understand, there is nothing in principle preventing my sequence instances from being NP-complete.