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We compare the performance of the popular Bondi-Hoyle-Lyttleton (BHL)accretion scheme with a simple mass-flux scheme applied to stellar-mass blackholes (BHs) across six levels of increasing spatial resolution. Simulating theformation of black holes within cosmological mini-haloes at $z \sim 20$, weinvestigate both the core-collapse and direct-collapse channels, which resultin BHs of initial mass $10.8 \, \text{M}_\odot$ and $270 \, \text{M}_\odot$respectively. Our explicit focus on the stellar-mass range pushes the maximumresolution down to sub-$10^{-3} \, \text{pc}$ regimes, where more complicatedgas dynamics are resolved. We observe efficient growth and rotationallysupported, $\sim$$10^{-1} \, \text{pc}$-scale discs around all $270 \,\text{M}_\odot$ BHs independent of resolution and accretion scheme, thoughclumps, bars, and spiral arm structures impact stability at high resolution. Weanalyse the effect of these instabilities on the accretion cycle. In contrast,all bar one of the $10.8 \, \text{M}_\odot$ BHs fail to attract a disc andexperience modest growth, even when characteristic scales of accretion anddynamical friction are reasonably resolved. While the two accretion schemessomewhat converge in mass growth for the $270 \, \text{M}_\odot$ case over $1\, \text{Myr}$, the greater degree of gas fragmentation induces more randomnessin the evolution of the $10.8 \, \text{M}_\odot$ BHs. We conclude that earlyuniverse black holes of $M_{\text{BH}} \sim 10^1 \, \text{M}_\odot$ struggle togrow even in gas-rich environments without feedback in comparison to seeds of$M_{\text{BH}} \sim 10^2 \, \text{M}_\odot$, and the latter exhibit convergentgrowth histories across accretion schemes below a spatial resolution of $1\times 10^{-3} \, \text{pc}$.

Hungry or Not: How Stellar-Mass Black Holes Grow (or Don't) in Dark Matter Mini-Haloes at High-Resolution