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.