Transformers are always the rub in designing switching inverters. If you notice the LM2577 notes at the beginning calls out a Ferroxcube core where you wind you own transformer.
Rewind--Looking at the Coilcraft website I found a transformer that should work for you
Coilcraft Q4438-BL Surface Mount Flyback Regulator Transformer. You would connect both output windings in series to get 24V.
It shows a different IC (LM258X)
**broken link removed**. This ap note
https://www.electro-tech-online.com/custompdfs/2010/01/AN-1095-1.pdf discusses using it as an isolated converter (Figure 6). It's actually a simpler circuit than the one using the LM2577
To get 24V output or so you would need to change the LM3411 regulator. Since you are not overly concerned about regulation accuracy I would try substituting a 22V zener diode in series with the output voltage and Ro, to replace the LM3411.
You may have to play with the value of Ro and perhaps add additional compensation as shown in Figure 14 to improve the converter stability.
P.S. Why do you need an isolated output?