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At this point, I feel like I've more or less run out of road, but was wondering if there was some way to force NDSolve to do what I want.

At this point, I feel like I've more or less run out of road, but was wondering if there was some way to force NDSolve to do what I want. UPDATE: I did try the model again with a different set of initial conditions, ones which allow for consistency between the boundary and initial conditions from $ t = 0 $ on, as shown below:

simpleLkInit = With[{m = 1/80},
   u0[a_] :> Piecewise[{{c - m*a, 0 <= a <= c/m}}, 0]];

simpleLkNormalization = First@Solve[
   {simpleLkIntegral2 /. t -> 0 /. u[a_, 0] :> u0[a] /. simpleLkInit, 
    c > 0}, c]
(* c -> 65/96 *)

Plugging this into NDSolve gives the same issue with overdetermination (so presumably the consistency of the boundary condition is never even checked):

NDSolve[{lkPDE /. \[Mu][_] -> 1/80 /. simpleLkInit /. 
   simpleLkNormalization, simpleLkIntegral2}, u, {a, 0, 100}, {t, 0, 
  100}]
(* Unevaluated, with NDSolve::overdet message *)

While the strategy of discretizing the system in age manually, as in Chris K's fine answer, is entirely viable, this essentially boils down to using the method of lines, which is the approach NDSolve itself uses. I would like to see if NDSolve itself can do the discretization, or at least if I can use it to drive move of the problem.