The gleim aviation weather book is a good level of meteorology knowledge for starting out in dispatch. Like others said you are bound by what the NWS /WSI is forecasting with a TAF (unless you have EWINS) and also the SOPs indicated by your company, so a great deal of forecasting knowledge and experience is not necessary; is it helpful, yes, but not essential. If you can read a TAF and understand how it relates with the overall pattern you are doing very well. I am a degreed met who forecasted for a while and IMO what they teach in dispatch school is sufficient. Forecasting/nowcasting is so much more learning from experience and practice rather than from a book or school. Some of the more seasoned dispatchers I work with seem to have a better grasp on meteorology applications than some of my former coworkers in av forecasting; sad but true. I wouldn't sweat it if you are just starting out.
For those looking for extra training with case study like examples I highly recommend this site:
https://www.meted.ucar.edu/training_detail.php
Some of the material is a bit complex, but there are plenty of courses that would be beneficial to a licensed dispatcher. There is also a whole section devoted to aviation and radar meteorology. It lists a skill level, I would avoid 3 and above which is grad level met material; even 2 can get a bit complex, junior/senior level meteorology. Completely free, just need a log-in. PM me you have any questions.
Cheers to WSI's excessive use of TEMPO and PROB!