How to fit piecewise functions with unknown domain?

Question

Thank you for your comments, I will discribe the question in more detail.

I have a piecewise function as below and I'm looking for the parameters :J0, C1, C2, a, Rp, R0, Z0, K, \[Delta], A

    J[R_, Z_, J0_, C1_, C2_, a_, Rp_, R0_, Z0_, K_, \[Delta]_, A_] = Piecewise[{
{J0 (1 - \[Rho]s[R, Z, a, Rp, R0, Z0, K, \[Delta], A]^2)^A, 0 <= \[Rho]s <= 0.7},
{J0 (C1 (1 - \[Rho]s[R, Z, a, Rp, R0, Z0, K, \[Delta], A])^2 + C2 (1 - \[Rho]s[R, Z, a, Rp, R0, Z0, K, \[Delta], A])^3), 0.7 < \[Rho]s <= 1},
{0, 1 < \[Rho]s}
    }];

And \[Rho]s[R, Z, a, Rp, R0, Z0, K, \[Delta], A] is shown as below:

x[R_, a_, Rp_] := (R - Rp)/a;
y[Z_, a_, Z0_] := (Z - Z0)/a;
\[Zeta][Z_, a_, Z0_, K_] := y[Z, a, Z0]/K;
\[Sigma][a_, Rp_, R0_] := (Rp - R0)/a;
\[Rho]s[R_, Z_, a_, Rp_, R0_, Z0_, K_, \[Delta]_, A_] :=
   1/(1 - \[Sigma][a, Rp, R0]^2) (((x[R, a, Rp] + \[Delta] \[Zeta][Z, a, 
           Z0, K]^2)^2 + (1 - A^2) \[Zeta][Z, a, 
         Z0, K]^2)^0.5 + \[Sigma][a, Rp, 
       R0] (x[R, a, Rp] + \[Delta] \[Zeta][Z, a, Z0, K]^2));

Then I want to time another function M and integrate it in range {R, 0.2, 1, d}, {Z, -1, 1, d} . Here d was set as 0.05 to reduce the amount of calculation.

Therefore, the code is

   Msum = 0;
   Do[{
   Msum = J[R, Z, J0, C1, C2, a, Rp, R0, Z0, K, \[Delta], A] d^2 CP[Rphi, Zphi, R, Z] + Msum;   
   }, {R, 0.2, 1, d}, {Z, -1, 1, d}];

Here the function CP[Rphi, Zphi, R, Z] is the response function of an unit Helmholtz current:

   CP[Rphi_, Zphi_, R_, Z_]:= 4 Pi 10^-7 (( 2 R Rphi)/Sqrt[(Zphi - Z)^2 + 
   (Rphi + R)^2](2/((4 R Rphi)/((Zphi - Z)^2 + (Rphi + R)^2)) 
   (EllipticK[((4 R Rphi)/((Zphi - Z)^2 + (Rphi + R)^2))] -  
   EllipticE[((4 R Rphi)/((Zphi - Z)^2 + (Rphi + R)^2))]) - 
   EllipticK[((4 R Rphi)/((Zphi - Z)^2 + (Rphi + R)^2))]));

Then the final function Msum is used for fitting the measured data. The measureddata, Rphi and Zphi is shown below.

Rphi https://drive.google.com/file/d/1PUB1A9ap_MqztbLtWcYCsTjrhuIjKuIA/view?usp=sharing

Zphi https://drive.google.com/file/d/1FXYvlQI2tYHwKfw2Xm_bV5cRiBHf_wOO/view?usp=sharing

measureddata https://drive.google.com/file/d/1I6E6vZhVUuH0l9pjltK0-LwGCij8-g8N/view?usp=sharing

I tried to use the FindFit :

data = Table[i + j, {i, 1, 64}, {j, 1, 3}];
Do[{
  data[[n, 1]] = Rphi[[n]] 10^-3; data[[n, 2]] = Zphi[[n]] 10^-3;  
  data[[n, 3]] = measureddata[[n]];
  }, {n, 1, 64}];
model = Msum;
FindFit[data, model, {J0, C1, C2, a, Rp, R0, Z0, K, \[Delta], A}, {Rphi, Zphi}

But it didn't work. Only FindFit[...1...] was shown in Out.

Then I tried to reduce number of parameters that only A is unknown. Additionally, the piecewise function was also simplified.

J0 = -500;
K = 1; 
\[Delta] = 0; 
Rp = 0.5;  
Z0 = 0.17; 
R0 = 0.6;  
a = 0.2;
J[R_, Z_, A_] = Piecewise[{
    {J0 (1 - \[Rho]s[R, Z, a, Rp, R0, Z0, K, \[Delta], A]^2)^A, 0 <= \[Rho]s <= 1},
    {0, 1 < \[Rho]s}
    }];

Here \[Delta] is ellipticity, hence the figure of \[Rho]s should be an eccentric circle.

However, FindFit:The Jacobian is not a matrix of real numbers at {A} = {1.}` was get.

What I'm confused is that the magnetidue of function\[Rho]s is decided on the parameters I'm looking for. I have no idea about how to conduct the fitting using such a piecewise function.

It will be appreciated if someone can help to improve this code. If the calculation take times you can reduce the number of parameters.