Quick reply while I have a sec... I had a bit of a learning curve when I first started using a shotgun mic on my Sony ZV-E10. Honestly, I was obsessed with the signal-to-noise ratio and found that you really have to be careful with how the camera handles preamps. I ended up with a setup where the mic has its own internal battery power because relying on the camera for juice sometimes introduced a weird floor noise in my indoor talking head segments. Make sure to look out for these technical bits:

• High-pass filters are basically essential for cutting out wind or traffic rumble outdoors.
• Manual gain control on the mic itself helps keep the camera's noisy preamps low.
• Cable shielding matters more than you'd think for preventing interference buzz. I learned the hard way that a mic might look good on paper but if it doesn't have a decent shock mount, your b-roll will be ruined by handling noise. My current setup uses a -10dB pad which saved my audio more than once when things got loud unexpectedly, so maybe keep an eye out for that feature.

For price around $200, I suggest you get the Sennheiser MKE 400: https://www.bhphotovideo.com/c/search?Ntt=Sennheiser+MKE+400+Shotgun+Microphone&BI=8941&KBID=10361&SID=12345&DFF=d50

Some Rode microphones are great for ZV-E10.

I think DJI MIC 2 is the best choice, though a bit expensive than your budget.