We perform the first three-dimensional radiation hydrodynamical simulations that investigate the growth of intermediate-mass BHs (IMBHs) embedded in massive self-gravitating, dusty nuclear accretion disks. We explore the dependence of mass accretion efficiency on the gas metallicity $Z$ and mass injection at super-Eddington accretion rates from the outer galactic disk $dot{M}_{rm in}$, and find that the central BH can be fed at rates exceeding the Eddington rate only when the dusty disk becomes sufficiently optically thick to ionizing radiation. In this case, mass outflows from the disk owing to photoevaporation is suppressed and thus a large fraction ($gtrsim 40%$) of the mass injection rate can feed the central BH. The conditions are expressed as $dot{M}_{rm in} > 2.2times 10^{-1}~M_odot ~{rm yr}^{-1} (1+Z/10^{-2}~Z_odot)^{-1}(c_{rm s}/10~{rm km~s}^{-1})$, where $c_{rm s}$ is the sound speed in the gaseous disk. With increasing numerical resolution, vigorous disk fragmentation reduces the disk surface density and dynamical heating by formed clumps makes the disk thickness higher. As a result, the photoevaorative mass-loss rate rises and thus the critical injection rate increases for fixed metallicity. This process enables super-Eddington growth of BHs until the BH mass reaches $M_{rm BH} sim 10^{7-8}~M_odot$, depending on the properties of the host dark-matter halo and metal-enrichment history. In the assembly of protogalaxies, seed BHs that form in overdense regions with a mass variance of 3-4$sigma$ at $zsim 15-20$ are able to undergo short periods of their rapid growth and transits into the Eddington-limited growth phase afterwards to be supermassive BHs observed at $z>6-7$.
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