Welcome aboard! I don’t have personal experience with any of the headphones you’ve listed, but we have a thread devoted to TYGR vs MMX 300 (although it quickly devolves into a lot of unrelated stuff, as usual).
The K361 and K371 have extra bass but unusually come back to neutral before the mid-range frequencies start. However, their build quality has been repeatedly reviewed as a bit iffy.
Another headphone I’d add is the new and hard-to-find-in-stock Sennheiser HD 560S (of course, we also have a thread on it). Early returns suggest it should be fine for gaming but should be especially appropriate for classical and pop due to its neutral sound signature.
While I don’t know anything about your particular desktop sound set-up, in general desktops have a problem with insufficiently shielded electrical interference between the various components diminishing sound quality. The many gaming enthusiasts on this forum may well know whether that would be true for your particular set-up. If so, you might be better allocating part of your budget to even a modest off-board amp+DAC unit and a slightly lower priced headphone, yet still end up with better sound quality.