Direct collapse models for black hole (BH) formation predict massive ($sim 10^5 M_{odot}$) seeds, which hold great appeal as a means to rapidly grow the observed $sim 10^9 M_{odot}$ quasars by $zgtrsim 7$; however, their formation requires fine-tuned conditions. In this work, we use cosmological zoom simulations to study systematically the impact of requiring: 1) low gas angular momentum, and 2) a minimum incident Lyman Werner (LW) flux radiation in order to form direct-collapse BH seeds. We start with a baseline model (introduced in Bhowmick et. al. 2021) that restricts black hole seed formation (with seed masses of $M_{mathrm{seed}}=1.25times10^{4},1times10^{5} & 8times10^{5}M_{odot}/h$) to occur only in haloes with a minimum total mass ($3000times M_{mathrm{seed}}$) and star forming, metal poor gas mass ($5times M_{mathrm{seed}}$). When seeding is further restricted to halos with low gas spins (i.e. smaller than the minimum value required for the gas disc to be gravitationally stable), the seeding frequency is suppressed by factors of $sim6$ compared to the baseline model regardless of the mass threshold used. In contrast, imposing a minimum LW flux ($>10J_{21}$) disproportionately suppresses seed formation in $lesssim10^9M_{odot}/h$ halos, by factors of $sim100$ compared to the baseline model. Very few BH merger events occur in the models with a LW flux criterion, and because early BH growth is dominated by mergers in our models, this results in only the most massive ($8times10^{5}M_{odot}/h$) seeds being able to grow to the supermassive regime ($gtrsim10^6M_{odot}/h$) by $z=7$. Our results therefore suggest that producing the bulk of the $zgtrsim7$ BH population requires alternate seeding channels, early BH growth dominated by rapid or super-eddington accretion, massive seeding scenarios that do not depend on LW flux, or a combination of these possibilities.
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