It is so funny that science never develops as in the textbooks. Bose only introduced his statistics in 1924, so Planck could not possibly have used it to derive the radiation formula in 1900. So how did he do it? Also, are there any translations of his original paper?
Bose derived the black body radiation formula in early 1924 by considering the ideal gas of light quanta. Nothing could be further from Planck's mind in 1900. The idea of light quanta did not appear until Einstein's 1905 photoeffect paper, in 1908 he thought of them as vortices in EM field rather than localized energy packets, and in 1911 remarked that the idea "does not seem reconcilable with the experimentally verified consequences of the wave theory". Planck, Nernst, Rubens and Warburg recommending Einstein's admission to the Prussian Academy of Sciences wrote:
"That he might sometimes have overshot his target in his speculations, as for example in his light quantum hypothesis, should not be counted against him too much".
Even after Bohr's 1913 model of the atom Einstein was one of very few who took light quanta seriously. See What was different about Planck's quantization of light compared to Einstein's?
Planck treated radiation classically, where statistics came in was in what emitted it. Like many physicists he was partly influenced by Mach's and Ostwald's skepticism about atoms, so he assumed that the radiation is produced by some ideal oscillators of unknown nature. In 1899 following the ideas of kinetic theory he found an expression for the entropy of his oscillators that reproduced Wien's phenomenological radiation law, but experiments quickly showed it to be incorrect at low frequencies.
Planck originally obtained the correct law in October 1900 by simply modifying his entropy expression, without any statistics. Two months later he was forced to turn to statistics to justify his entropy formula, and formulated the "Boltzman equation" $S=k\ln W$. To find the disorder $W$ he had to count the number of ways that a given energy can be distributed among the oscillators. It is for that that he needed "energy droplets", which were supposed to be fictitious and go away when the continuous limit was taken. But instead of taking the limit Planck was forced to introduce a "new constant of nature", to fix the amount of energy in the droplets, "a purely formal assumption and I really did not give it much thought except that no matter what the cost, I must bring about a positive result", as he wrote in 1931. See Kragh's Max Planck: the Reluctant Revolutionary.
Planck continued to not give it much thought for five years, and he certainly did not think that his counting was in conflict with classical physics. The source of discreteness was unclear, Lorentz, who was first to give it a thought in 1903, subsequently pondered various classical mechanisms (in interactions of the oscillators with ether, atoms, electrons, etc.). Rayleigh in June 1900 argued that equipartition of energy in classical statistical mechanics leads to a discrepancy with radiation experiments, and in a 1905 paper with Jeans derived a different radiation formula. As Kragh writes:
"The result is an energy density that keeps on increasing as the frequency gets higher and higher, becoming "catastrophic" in the ultraviolet region. In spite of its prominent role in physics textbooks, the formula played no part at all in the earliest phase of quantum theory. Planck did not accept the equipartition theorem as fundamental, and therefore ignored it".
The "ultraviolet catastrophe" only became an issue when Lorentz showed in 1908 that the Rayleigh-Jeans law obtains even under the most general (classical) assumptions about a system of atoms, electrons and ether. The nickname was introduced by Ehrenfest at Solvay in 1911. So much for textbooks. See Kox's Hendrik Antoon Lorentz’s Struggle with Quantum Theory.
One last bit of historical peculiarity. Bose was counting microstates in a funny way, which implied heavy statistical dependence between particles, apparently without realizing it. Einstein, to whom Bose sent his paper, did not notice the funny math either until Ehrenfest pointed it out to him later in 1924. Only in the January 1925 paper he finally acknowledged it explicitly with an "explanation" that it is "completely mysterious", but hey, it gives the right answer. Only after the wave and matrix mechanics were introduced a year later could the funny math be justified by indistinguishability of particles. See Delbruck's Was Bose-Einstein Arrived at by Serendipity?