I have run rack rigs. I am not necessarily opposed to the idea, but I think your vision may be a bit misguided. Your must haves have no mention of your artistic vision (the type of sound you want). The range of possible tones a preamp can produce depends on how it is voiced and how the EQ section operates, which can vary widely. For me the sound is usually the primary justification.
The current preamp rack preamp I thought of that meets the must haves is a Jule Monique. The "new price" is double your budget. This is a mono preamp, so it will not meet you channel switching requirements.
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I own the version of Monique with a built in 700W amp. It's a fabulous piece of kit. However some people love the way it sounds and some don't.
I am not familiar with Helix or other modeler options. However I know they are versatile. Even if a modeler rack is one channel, you should be able to run a reverse ABY to switch instruments and then select patches custom tuned for whatever instrument and sound you want. My guess is this option will best meet your needs.
My last touring rig was rack based. The front end of the rig was an Ashly MX508 rack mixer. This mixer is over your budget, but there are plenty of options that cost less. Some even have High Z inputs on a couple of channels. I chose this route to facilitate switching between fretted bass guitar, upright bass with a pickup, and electric upright bass. This MX508 does not have instrument level inputs. I used Shure A95U line matching transformers so I could plug into the mic inputs. This gave me gain and EQ for each bass. Also the mixer had a button to select between line and mic inputs. This line selector effectively served as a mute switch.
If you can let go of the rack mount requirement, you may find one of the 2-channel Grace preamps meets your needs.
https://gracedesign.com/products/instrument-preamplifiers The Felix is over your budget.
Another company to consider is Headway:
Pre-amps – Headway Music Audio
Also Radial:
PZ-Pro - Radial Engineering
These three pedal based preamp are really intended for acoustic instruments.
I used to fantasize about certain rack preamps. Two I still think about from time to time are the Kern IP777 and Midas XL42. These are rare and long out of production and I have passed up reasonable deals on both.