  Because early black holes (BHs) grew to $\sim10^{9} ~M_\odot$ in less than 1
Gyr of cosmic time, BH seeding models face stringent constraints. To
efficiently constrain the parameter space of possible seeding criteria, we
combine the advantages of the cosmological IllustrisTNG (TNG) simulations with
the flexibility of semi-analytic modeling. We identify TNG galaxies as BH
seeding sites based on various criteria including a minimum gas mass of
$10^7$-$10^9~M_\odot$, total host mass of $10^{8.5}$-$10^{10.5}~M_\odot$, and a
maximum gas metallicity of $0.01 - 0.1 ~Z_\odot$. Each potential host is
assigned a BH seed with a probability of $0.01 - 1$; these BHs are then traced
through the TNG galaxy merger tree. This approach improves upon the predictive
power of the simple TNG BH seeding prescription, especially in the low-mass
regime at high redshift, and it is readily adaptable to other cosmological
simulations. Most of our seed models predict $z\lesssim4$ BH mass densities
that are consistent with empirical data as well as the TNG BHs. However,
high-redshift BH number densities can differ by factors of $\sim$ 10 - 100
between models. In most models, $\lesssim10^5~M_\odot$ BHs substantially
outnumber heavier BHs at high redshifts. Mergers between such BHs are prime
targets for gravitational-wave detection with LISA. The $z=0$ BH mass densities
in most models agree well with observations, but our strictest seeding criteria
fail at high redshift. Our findings strongly motivate the need for better
empirical constraints on high-$z$ BHs, and they underscore the significance of
recent AGN discoveries with JWST.
