eqp=(2*a*e*M)/(r*(a^2 - 2*M*r + r^2)) - (L*(2*M - r))/(r*(a^2 - 2*M*r + r^2))
eqt=(-2*a*L*M)/(r*(a^2 - 2*M*r + r^2)) - (e*(-2*a^2*M - a^2*r - r^3))/(r*(a^2 - 2*M*r + r^2))
eqr=-1 + e^2 + (2*(-(a*e) + L)^2*M)/r^3 + (a^2*(-1 + e^2) - L^2)/r^2 + (2*M)/r

solve for r'=0 (i.e., V=0), V'=0, V''=0 --a circular, min/max V; V''=0 is where the orbit just becomes unstable

Solve[{eqr==0,D[eqr,r]==0,D[eqr,{r,2}]==0},{e,L,r}]
out=% /. {a-> .999999, M->1}
{e -> 0.586424, L -> 1.17307, r -> 1.0161}

eqt /. Join[out[[6]],{M->1,a->.999999}] 
Out[9]= 144.625

The Solve finds many marginal orbit solutions, out[[6]] is the physical one
eqt is dt/dtau, plug into that for result.
Remark: many more 9s are needed to get this to 10,000