What are the all possible ways to increase the precision of this numerical integral?

I have to evaluate the following code

me = SetPrecision(1, 50);
mp = SetPrecision(400, 50);
M = mp + me;
(Omega) = SetPrecision(1000, 50);
(Gamma) = SetPrecision(0.5*M*(Omega), 50);
{(Alpha), (Beta)} = 
  SetPrecision({0.8086371511120481`, 495.09178203818004`}, 50);
chvar((Alpha)_, (Beta)_, re_, rp_) := 
  E^(-(((me re + mp rp)^2 (Gamma))/(me + mp)^2) - (Alpha) RealAbs(
     re - rp) - (Beta) RealAbs(re - rp)^2);
overlap((Alpha)_, (Beta)_) := 
  NIntegrate(
   re^2 rp^2 (E^(-(((me re + mp rp)^2 (Gamma))/(me + 
          mp)^2) - (Alpha) RealAbs(re - rp) - (Beta) RealAbs(
         re - rp)^2))^2, {re, 0, 10}, {rp, 0, 10});
lapP((Alpha)_, (Beta)_, re_, rp_) := 
  E^(-(((me re + mp rp)^2 (Gamma))/(me + mp)^2) - (Alpha) RealAbs(
       re - rp) - (Beta) RealAbs(re - rp)^2) (-2 (Beta) - (
      2 mp^2 (Gamma))/(me + mp)^2 + ((re - rp)^2 (Alpha))/
      RealAbs(re - rp)^3 - (Alpha)/RealAbs(re - rp)) + (
   2 E^(-(((me re + mp rp)^2 (Gamma))/(me + 
        mp)^2) - (Alpha) RealAbs(re - rp) - (Beta) RealAbs(
       re - rp)^2) (2 (re - rp) (Beta) - (
      2 mp (me re + mp rp) (Gamma))/(me + 
        mp)^2 + ((re - rp) (Alpha))/RealAbs(re - rp)))/rp + 
   E^(-(((me re + mp rp)^2 (Gamma))/(me + mp)^2) - (Alpha) RealAbs(
       re - rp) - (Beta) RealAbs(re - rp)^2) (2 (re - rp) (Beta) - (
      2 mp (me re + mp rp) (Gamma))/(me + 
        mp)^2 + ((re - rp) (Alpha))/RealAbs(re - rp))^2;
kinP((Alpha)_, (Beta)_) := -1/(2* mp)*
   NIntegrate(
    re^2*rp^2*chvar((Alpha), (Beta), re, rp)*
     lapP((Alpha), (Beta), re, rp), {re, 0, 10}, {rp, 0, 10});
TP = kinP((Alpha), (Beta))/overlap((Alpha), (Beta))

Evaluating this code gives 749.661702852 for TP while I know the true value is 750.0221. Since obtained value is close to true one, seems that it’s just a matter of precision (also Mathematica gives warnings during evaluation), so I added

WorkingPrecision -> 60, MaxRecursion -> 50, PrecisionGoal -> 10

for two included NIntegrate commands in the code, but it didn’t improve the result as I expected, Any idea to get true result?
Any help would be appreciated