Is it possible to generalize Ramanujan's lower bound for factorials that he used in his proof of Bertrand's Postulate?

After this step: $$ \lfloor\frac{x}{b_1}\rfloor + 1 - \frac{x}{b_1} \ge (\lfloor\frac{x}{b_2}\rfloor + 1 - \frac{x}{b_2} - \frac{1}{2}) + (\lfloor\frac{x}{b_3}\rfloor + 1 - \frac{x}{b_3} - \frac{1}{2}) $$ we have $$ \lfloor\frac{x}{b_1}\rfloor + 1 \ge (\lfloor\frac{x}{b_2}\rfloor + 1) + (\lfloor\frac{x}{b_3}\rfloor + 1)-1 $$ but what you have is incorrect. For example, let $x=77.1,b_1=6,b_2=7,b_3=42$, then $\lfloor x/b_1\rfloor=12, \lfloor x/b_2\rfloor=11, \lfloor x/b_3 \rfloor=1$ and $13\not\ge 14$.

And in this case your inequality is also violated: $$ \ln \Gamma\left(\frac{77.1}{6}\right)-\ln\Gamma\left(\frac{77.1}{7}+\frac{1}{2}\right)-\ln \Gamma\left(\frac{77.1}{42}+\frac{1}{2}\right) = 3.10698\cdots \\ \ge \ln(12!)-\ln(11!)-\ln(1!) = 2.4849\cdots $$