I think I understand, you're talking about the numerical integration, right? It's probably something of an academic question because you can just set the error to 1e-9 or whatever, and the convergence of that approximation is so good it doesn't slow you down appreciably.
That said, there are other things you can do. Since the derivatives of an Euler spiral are easy enough to compute analytically, you could do Hermite interpolation with a polynomial of arbitrarily high order. I've worked out how to do it for quintic, which is pretty tractable, but I'm sure it could be done even higher. That definitely meets your stated goal (precise at endpoints, sloppy in the middle), but the order of convergence of a quintic is only moderately good compared to the higher order polynomials in spiro.
That said, there are other things you can do. Since the derivatives of an Euler spiral are easy enough to compute analytically, you could do Hermite interpolation with a polynomial of arbitrarily high order. I've worked out how to do it for quintic, which is pretty tractable, but I'm sure it could be done even higher. That definitely meets your stated goal (precise at endpoints, sloppy in the middle), but the order of convergence of a quintic is only moderately good compared to the higher order polynomials in spiro.