UPDATE!!! I found a fix! In GZDoom, I originally was loading the mod through Chex Quest, but instead what I did to fix the issue was that I selected Ultimate Doom, and then put "-file CHEX.WAD -file FURDoom_BETAv0.1.pk3" in the Aditional Parameters and surprisingly that fixed the weapon switching issue by replacing Chex Quest's Weapons ENTIRELY with FurDoom's, and also the Flemoids are replaced by the Furry Demons too. The only problem is that beating the final level takes you to Doom's E1M6 instead of finishing the game... Oh, well. I guess it's sort of a fun little gimmick anyway. ¯\_(ツ)_/¯