
Shure
SM58
Indestructible live workhorse; but muffled for studio vocals.

Running these analyses costs money. Buy through my links to help keep lights on! I may get a small commission.
Buy a used sennheiser mkh 416 p48 they can be had for 5-600!! Great microphone
It handles weather changes and humidity just like the 416, more forgiving, shorter and lighter, DPA sound.
I really will never understand the tlm103 love. For me, it’s u87 or MKH416 for that commercial cut-through.
I would cross post this on /locationsound. Anyways, for what you are looking to achieve I would find a pair of used lectro 411s, mkh416, zoom h8 or a mix pre. Since that is over your budget, I would buy what you can afford and rent till you can buy more gear.
Sennheiser MKH-416 gets used in sooooo many spots. Everyone knows it, everyone likes it, studios buy them by the case. You would be hard pressed to find an on-camera spot that isn't using at least one 416. What the U87 is to LDCs, the 416 is to shotgun mics. It's on the spec sheets all the time, and I don't think I've ever seen a request for a different shotgun.
Let me think about it, it's a very delicate matter. I think having a good preamp is essential for your purpose. If you get a poor preamp, your sound will be poor even with a good microphone. I can tell you that there are some good preamps out there, but they're very expensive, and if you add the cost of the preamp and the microphone, we're going beyond your budget. Examples: the Fostex Ar501, created for your purpose, or the mm1 sound device, but we're on another level. The sound device is very expensive and perhaps useless for your application. There are similar, inexpensive solutions, but I can't guarantee the performance. As for the microphone, I obviously recommend a Sennheiser MKH416, but we're out of your budget. I'd go with a Deity S Mic 3s, which is within your budget considering the cost of the preamp. That said, if you're looking for a quick and simple solution without too many hassles, an NTG4+ or MKE600 would be better.

Shure
SM58
Indestructible live workhorse; but muffled for studio vocals.

Shure
SM57-LC
Versatile for instruments; needs pop filter for vocals.

DJI
Mic 2
Unbeatable convenience, 32-bit float; but poor Bluetooth audio.

Samson Technologies
Q2U USB/XLR Dynamic Microphone with Accessories
Great value dual USB/XLR; recent price increase noted.

Shure
SM7B Vocal Dynamic Microphone
Professional vocal sound; but expensive, needs external amplifier.