= Solution
Normalize the <linear growth factor> to $D(0)=1$. The collapse overdensity at the collapse epoch is $\delta_{\rm sc}\simeq1.686$ for the matter-dominated <spherical-collapse model>. The <present-extrapolated spherical-collapse barrier> is
$$
\boxed{\delta_c(t)=\frac{\delta_{\rm sc}}{D(t)}.}
$$
It is the initial linear overdensity, extrapolated to today, required to collapse by $t$; it is not the nonlinear <density contrast> of a virialized halo. The <smoothed matter density variance> $\sigma^2(M,z)$ is the variance of the linear <density contrast> smoothed on a Lagrangian comoving scale containing mass $M$. For a spherical <top-hat filter>,
$$
M=\frac{4\pi}3\bar\rho_{m,0}R^3,\qquad
\sigma^2(M,z)=\frac1{2\pi^2}\int_0^\infty k^2P(k,z)|W(kR)|^2dk,
\quad W(x)=3\frac{\sin x-x\cos x}{x^3}.
$$
Consequently $\sigma(M,z)=D(z)\sigma(M,0)$ for scale-independent linear growth. The equivalent <halo peak height> conventions are $\nu=\delta_c(t)/\sigma(M,0)=\delta_{\rm sc}/\sigma(M,z)$; using both an evolved barrier and an evolved variance would count growth twice.
To calculate a number-density growth time, use a narrow fixed-mass bin, not the collapsed mass fraction itself. Differentiate the <Press-Schechter formalism> mass fraction and divide the mass density in the resulting interval by $M$. This gives the <Press-Schechter halo mass function>
$$
\frac{dn}{dM}=\sqrt{\frac2\pi}\frac{\bar\rho_{m,0}}{M^2}
\left|\frac{d\log\sigma(M,0)}{d\log M}\right|\nu e^{-\nu^2/2}.
$$
At fixed $M$, the mass factor and logarithmic slope do not depend on time. Since $\dot\nu/\nu=-\dot D/D$, the <Press-Schechter abundance growth at fixed mass> is
$$
\boxed{\frac{\partial\log n}{\partial t}\bigg|_M=(\nu^2-1)\frac{\dot D}{D},\qquad
t_d=\left[(\nu^2-1)\frac{\dot D}{D}\right]^{-1}.}
$$
The prefactor $\nu$ matters here: the exponential-only rare-peak approximation drops the minus one and is accurate only for $\nu\gg1$.
Read the supplied variance relation as a mass-scale calibration using the numerical velocity label given for the selected population. It gives $\sigma(M,3)=0.85\sqrt{250/10}=4.25$. Matter domination between the two high-redshift epochs gives $D(19)/D(3)\simeq4/20$, so $\sigma(M,19)\simeq0.85$ and $\nu\simeq1.984$. At the epoch in question,
$$
H(19)=H_0\sqrt{0.25(20)^3+0.75},\qquad
H^{-1}(19)\simeq3.19\times10^8\,\mathrm{yr},\qquad \dot D/D\simeq H.
$$
Thus the requested fixed-mass-bin growth estimate with that calibration is
$$
\boxed{t_d\simeq\frac{3.19\times10^8}{(1.686/0.85)^2-1}\,\mathrm{yr}
\simeq1.09\times10^8\,\mathrm{yr}.}
$$
An exponential-only approximation gives about $8.1\times10^7\,\mathrm{yr}$, which is somewhat shorter because this is only a roughly two-sigma population.
The wording leaves two sample conventions worth distinguishing. First, an actual <virial velocity of a spherical-overdensity halo> is epoch-dependent at fixed mass. If the variance fit is instead calibrated using physical virial velocities at redshift three, the same mass whose velocity is $10\,\mathrm{km\,s^{-1}}$ at redshift nineteen has $v(3)\simeq10\sqrt{4/20}=4.47\,\mathrm{km\,s^{-1}}$. The <halo virial-velocity conversion between epochs> then gives $\sigma(M,19)\simeq1.271$, $\nu\simeq1.327$ and \b[$t_d\simeq4.2\times10^8\,\mathrm{yr}$] for a fixed-mass bin. The two numerical answers reflect the velocity-label convention in the supplied fit, not two ways of differentiating one fixed fit.
Second, a cumulative number density is $n(>M)=\int_M^\infty(dn/dM')dM'$, not simply $\bar\rho_{m,0}f(>M)/M$: the latter is a mass fraction divided by a threshold mass, not the number of objects. The cumulative derivative is an abundance-weighted average of $(\nu(M')^2-1)\dot D/D$ over that integral. If the supplied power-law variance fit is extended over all larger masses, with the same velocity-label convention, direct integration gives a cumulative-number growth time of about $8.4\times10^7\,\mathrm{yr}$ instead. A sample maintained at fixed physical velocity at successive epochs additionally moves its mass boundary and needs a selection convention. The numerical estimate above explicitly uses the ordinary fixed-mass differential interpretation. These distinctions are important when an exact growth time rather than a rare-tail estimate is intended.
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