Amazon Music High Definition and Ultra High Definition Equipment Requirements

My understanding is that streaming through an iPhone is capped at 24-bit/48 kHz lossless audio. Any DAC induced further upscaling from that - even if it is even possible with your suggested budget Schitt which I tentatively highly doubt - … is a big performance compromise at a minimum that won’t get you anywhere near the audio performance of a native hi-rez 24/192 pair of discrete hi-end units.

Toddle down to your fave bricks and mortar high-end audio store and actually go audition a dedicated high-end streamer capable of 24/192 as a source, now feeding into a high-end DAC capable of 24/192+ rez …. The audio performance improvement provided by each component in the link are huge.

Amazon Music is a little more difficult to stream than some of the other services.  They don't share the API and don't support Linux like some of the others. 

An Amazon Echo Link  is one solution.  it does have cozx and toslink out to connect to your dac or RCA out to bypass it

Personally I use an inexpensive i5 Fanless PC and USB to my DAC.  There are lots of RemotePC aps to control the computer from your phone.  This gave me the highest resolution for not much more than a Link

I used to stream from my iPhone, but using a separate streamer sounds much better, but I don’t think too many streamers work with Amazon HD. I’d suggest eventually stepping up to using a streamer and switching to Qobuz that reportedly sounds better than Amazon HD. But, if you’re gonna go from your phone I’d highly suggest ditching the Apple Camera Adapter and get this LavriCables adapter as it only costs $85 and sounds much better.
https://www.lavricables.com/cables/reference-silver-lightning-interconnect-cable-for-oppo-ha-2-centrance-mini-m8/

You might also consider their dual USB cable that uses separate signal and data legs for better sound and that is usually reserved for much pricier cables.

https://www.lavricables.com/cables/ultimate-silver-dual-usb-a-b-interconnect-cable/

I have both of these and other LavriCables and they’re great. Hope this helps, and best of luck.

I just got a Wiim Mini today, streaming Amazon HD via toslink to my preamp DAC and working great.  You can control either via the wiim app or stream via the Amazon HD app.  $84.00 on Amazon.

https://wiimhome.com/WiiMMini/Overview

Get a DTS Play-Fi streaming device (Paradigm Link, Martin Logan Unison, Klipsch Gate, Audiolab 9000N, etc) and you load the Play-Fi app on your phone, hook the streamer up to your system and you can stream hirez from amazon HD or many other services. prices range from $30 to $2000 depending on the device you choose. The ML Unison also does airplay, works with Alexa, has ARC room correction and is 50% off:

https://www.martinlogan.com/en/product/unison