No offense meant here but you got a nice mic and want to butcher it now? Look at Rycote. They make blimps and fur that work and have a size to fit about anything. You sound will only be as good as your weakest link. You get what what you pay for.
The small cost of batteries is worth it. When wiring the cast for a live theatre performance that sometimes uses 25+ wireless beltpack transmitters. Before each show each one gets a new battery AND after the show they are left on intentionaly to go completely dead. That way they is never any doubt if the battery is new or old , good or bad.
Suggest you rent sommething,look at Lectrosonics. Don’t waste your money on a cheap mic system, don’t expect quality audio for this price. Think about it , you spent about $4000 for your camera, right? don’t plug garbage into it
Have you viewed the difference in quality between a 1/3 chip or 1/2 or 2/3 chip camera. 150 and 170 may be nice but can’t stand up to a camera with a real lens