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Without diving too much into detail... Basically, you will definitely experience an improvement with a better mouse. However, if you're expecting to go from bronze to Taimou because of a mouse, you're probably going to be disappointed.
I would recommend the following as an extremely reasonably priced, respectably performing gaming mouse: Corsair Harpoon -
Pixart PMW3320 Sensor, 250-6000 DPI, 1000Hz polling
I don't really have any comments on the headset or keyboard, as they seem like reasonable choices. Though I have no experience with HyperX or Cherry as a KB manufacturer.
General thoughts on a headset though, go wireless if you can. It's SOOO much nicer than a wired set.
As for the mouse... I have never even heard of Zowie. That mouse looks extremely basic for $60. If you don't want any fancy features in a mouse, why not go cheaper there? Here's an option for $20 from Corsair Most major manufacturers have options for less than $60 with more or different features.
The Corsair Harpoon is a good small mouse with RGB lighting placed on RocketJumpNinja's top 40 list.
Anyone have experience w this mouse? Considering snagging it if it's a good budget gaming mouse.
Get this mouse and you're good to go: https://www.amazon.com/dp/B01M9DDCKD
I argue this really isn't a good deal particular to Best Buy. Amazon 3rd party sellers have it for slightly less and eBay has one, BNIB, for $15 shipped/excl. tax.
I know you said the majority of the budget should go towards the mouse, but you think this is good?
Or
The Corsair harpoon is a good choice that is a bit under $30.
Corsair Harpoon. 4.4 x 2.7 x 1.6 in. I tried it and it felt really well-built. No sensor rattle and solid. I didn't find it comfortable, however, particularity on the right side, but then again I use a weird grip. My ring finger presses up high against the right side and this mouse is not meant for that. It may be fine for a palm grip. Rocket ninja review https://www.youtube.com/watch?v=PDUZoecJIjU
I Reccomend a new mouse and keyboard. Good budget ones are: