2-SAT reduction to Easy Puzzle Solving
I'm going to need to question your premise here.

I have no doubt that you can poly-time reduce 2SAT to your custom problem. But you can poly-time reduce 2SAT to pretty much anything. The idea of "P-hardness" under poly-time reduction isn't really a useful one.

For example, let's take the following problem:


  IS-IT-ONE: given a single bit, is that bit a 1?


This problem is "P-hard" under poly-time reduction. For example, I can reduce 2SAT to it:


  Reduction from 2SAT to IS-IT-ONE
  
  
  Solve 2SAT the usual way
  If the problem is satisfiable, let $B$ be 1
  If the problem is unsatisfiable, let $B$ be 0
  Run IS-IT-ONE on $B$ and return the result
  


This reduction runs in poly-time, because solving 2SAT can be done in poly-time.

In other words:

Every problem in P can be poly-time reduced to any other non-trivial problem.

(Only two languages are considered "trivial" for this purpose: "return true" and "return false", aka $\Sigma^*$ and $\varnothing$.)

If you're interested in classes smaller than P, great! There are several of them, such as L: the set of problems that can be solved using a logarithmic amount of memory. It may or may not be the same as P—that's an open problem.

But if you want to talk about P versus L or NL or anything else within P, you can't use poly-time reductions any more. When talking about L, for example, you'll usually want a log-space reduction, using a Turing machine with writable tape bounded by the log of the input size. These aren't as famous as poly-time reductions, but they're still quite important in complexity theory.