This is off the top of my head, so it may be total bullshit. I find the idea of memory in a timeless universe slippery myself, and can only occasionally believe I understand it. But anyway...

If you want to implement a sort of memory in your 2D space with one particle, then for each point (x0,y0) in space you can add a coordinate n(x0,y0), and a differential relation

dn(x0,y0) = δ(x-x0,y-y0) sqrt(dx^2 + dy^2)

where δ is the Dirac delta. Each n(x0,y0) can be thought of as an observer at the point (x0,y0), counting the number of times the particle passes through. There is no reference to a time parameter in this equation, and yet there is a definite direction-of-time, because by moving the particle along a path you can only increase all n(x0,y0) for points (x0,y0) along that path.

A point in this configuration space consists of a "current" point (x,y), along with a local history at each point. If you don't make any other requirements, these local histories won't give you a unique global history, because the points could have been visited in any order. But if you impose smoothness requirements on x and y, and your local histories are consistent with those smoothness requirements, then you will have only one possible global history, or at most a finite number.