Question: LSSolve on a large ODE system

hello i have the following set of ode's:

ode_sub := diff(S(t), t) = -k1*S(t)-S(t)/T1_s;
ode_P1 := diff(P1(t), t) = k1*S(t)-k2*(P1(t)-P2(t)/keq)-P1(t)/T1_p1;
ode_P2 := diff(P2(t), t) = -k2*(-keq*P1(t)+P2(t))/keq-k4*P2(t)-P2(t)/T1_p2;
ode_P2e := diff(P2_e(t), t) = k4*P2(t)-P2_e(t)/T1_p2_e;

ode_system := ode_sub, ode_P1, ode_P2, ode_P2e;

with these parameters:
s0 := 10000;
k2 := 1000; T1_s := 14; T1_p2_e := 35; T1_p2 := T1_p1;
 

i want to find the unkown parameters : T1_p1, k1, keq and k4

my idea was this:

init:=S(0)=s0,P1(0)=0,P2(0)=0,P2_e(0)=0

dsolve({ode_system,init})

sol := combine(expand(%));
PS := subs(sol, [S(t), P1(t), P2(t), P2_e(t)]);
 

P1fu := unapply(PS[2],t);
Sfu := unapply(PS[1],t);
P2fu := unapply(PS[3],t);
P2e_fu := unapply(PS[4],t);
P2_total := unapply(P2fu+P2e_fu, t);
 

the following data is given:

T:=<0,2,4,6,8>

S:=<9999.99913146527,8328.870587730016,6937.009129218748,5777.745632133724,4812.209983843559>

P1:=<0.0,67.86790056712294,114.88787098501874,145.95438088662502,164.85650644237887>

P2_P2e:=<0.0,271.68492651947497,461.9130396605823,589.3710176125417,668.9967533337124> # data from P2(t)+P2_e(t)

making the rediduals:

RP1 := convert(P1-P1fu~(T), list);
RS := convert(S-Sfu~(T), list);
RP2_P2e := convert(P2_P2_e-P2_total~(T), list);
 

RPs := [op(RS), op(RP2_P2_e), op(RP1)]

res := Optimization:-LSSolve(RPs, k1 = 0 .. 1, keq = 0 .. 10, k4 = 0 .. 1, T1_p1 = 0 .. 100)

i dont know wheter or not the last step work to get the parameters becuase it takes to long to compute. is there a smarter way to obtain the parameters of the ode's? a numeric approch ?

i tried with dsolve({ode_sysytem,init},numeric,'parameters'=[k1,keq,k4,T1_p1]) however it doesnt seem to get my anywhere since i need to know the parameters to use this (i think)

hope someone can help:)