Improving speed performance of 'Fit' with BSplines?

Question

Context

In the context of this (vaguely controversial) question, I am stumbling into a performance issue with Fit.

Let me illustrate on fitting a set of BSplines to set of 64 data points:

dat = Table[{i, Sin[i^2]}, {i, 0, 2 Pi, Pi/32}] // N;
fspl = Interpolation[dat, Method -> "Spline"];
fb = First[Cases[fspl, _BSplineFunction, ∞]];
{sd, scpts, sk} = fb /@ {"Degree", "ControlPoints", "Knots"};
func = Flatten[Outer[Times, Sequence @@ MapThread[
Table[BSplineBasis[{#1, #2},k-1, #3], {k, #4}] &, {sd, sk, {x}, Dimensions[scpts]}]]];
fit = Fit[dat, func,x, FitRegularization -> {"Curvature", 1.}]; // Timing

(* 0.008626 *)

Plot[fit, {x, 0, 2 Pi}]

But if I now ask say for 1024 points

dat = Table[{i, Sin[i^2]}, {i, 0, 2 Pi, Pi/512}] // N;
fspl = Interpolation[dat, Method -> "Spline"];
fb = First[Cases[fspl, _BSplineFunction, ∞]];
{sd, scpts, sk} = fb /@ {"Degree", "ControlPoints", "Knots"};
func = Flatten[Outer[Times, Sequence @@ MapThread[
Table[BSplineBasis[{#1, #2}, k-1, #3], {k, #4}] &, {sd, sk, {x}, Dimensions[scpts]}]]];
fit = Fit[dat, func,x, FitRegularization -> {"Curvature", 1.}]; // Timing

(* 5.83293 *)

It takes dramatically much longer! (and it get much worse with higher sampling)

Question

Is is possible to improve the performance of Fit on this specific problem?

Thank you for your help.

I am aware that I could ask for less BSplines but...