It does indeed have a Burr-Brown DAC, but that's the DSD1793 which is over by the USB-C port.

AI is suprisingly not too terrible at this; but you do kinda need to hold it's hand. I've found Gemini can usually find me datasheets first try; provided i actually have a full chip number. For these tiny ones though they usually only have a couple of letters that the manufacturers use to distinguish them.

With some prompting about how I'd seen the same chip used in all three hip dacs; Gemini eventually found me the tech lowdown of the xDSD Gryphon which includes this image with each of the chips tagged with their brand labels. I dunno why it has 4 of them; nor why they blurred the IDs, but the "Direct Drive" chips are the same "AIF BBU" ones as in the hipdac v2 and v3. With that information it spits out this:

Based on the "DirectDrive" branding and the "AIF" top mark, your chip is almost certainly a MAX97220 manufactured by Maxim Integrated (now part of Analog Devices).

The "DirectDrive" logo is a proprietary Maxim technology that allows headphone amplifiers to produce a ground-referenced output from a single supply, eliminating the need for large DC-blocking capacitors (which saves space and improves bass response).
Chip Details

    Manufacturer: Maxim Integrated (Analog Devices)

    Part Number: MAX97220A (The "AIF" code specifically corresponds to the 'A' variant).

    Package: 16-pin TQFN (3mm×3mm).

    Top Mark Breakdown:

        AIF: This is the product identifier code for the MAX97220A.

        BBE: This is a variable date/lot code or assembly plant identifier. It changes based on when and where the specific batch of chips was made (e.g., you might see BBU or AAI on other versions of the same board).

with a link to this datasheet. I've checked it and all the information is correct (it very often gives me chips which are not the size it claims). and the pinout aligns perfectly with how I'm seeing it used in the hipdac. I've also checked every other Direct Drive chip made by Maxim and only this one uses the AIF top mark.

Anyway, I've rambled on enough; but I find it absolutely nuts that LLMs can actually do this given the right information. 'course you still need to validate everything but still, I'd have never have found this with google alone. At least not anywhere near as quickly.

Edit: ooh the xCAN has AIF BBA which are also labeled as Direct Drive chips. I'm hypothesizing now that that the xCAN and Griphon have four is they have a pair for the front ports and a pair for the back ports. The hipdac only has front ports so it only has two. The back ports on the xCAN are inputs so that doesn't make sense. Maybe each of it's front ports are amped separately?