@ Maniac Sorry for the late answer, I was on the road. Hope it is not too late for you.
If think it’s a pity to feed the Hecate 5V and the 3,3V module with the same PSU. This because, the Hecate board put in the game many care to isolate its output (I2S) from its Input (USB). They use some top notch galvanic (magnetic cells) isolation and a full dedicated 3,3V. The small black box on the board produce a 3,3V from the 5V with a separated ground ((it seems to be this component). All numeric signals are isolated too. Thus is you re-use the same 5V PSU to power both the USB 5V of the Hecate and the I2S/HDMI module (even through the 3.3V low noise regulator), you’ll share the same ground for both. The HDMI module must be grounded with the 3,3V of the I2S output of Hecate. Don’t know if I’m clear, sorry.
If you can, use your 1,5A Linear PSU to power only the Hecate board on its USB input and another 5V PSU (100mA is sufficient) to feed the 3,3V LT3045 you bought. The LT3045 accepts up to 20V. The LT3045 is a Low Voltage Drop Out, only 300 or 500mV, so a 5V in input should be sufficient.
Here are the options, you have, and, to understand, keep in mind the I2S ground of the DirectStream input is connected to the I2S/LVDS ground of the HDMI module, then to the I2S ground of the Hecate output:
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Re-use the 5V PSU you use for the Hecate Input. As just said before, you’ll link all the grounds everywhere, so, you lose all the benefits of the isolation implemented on the Hecate board.
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Put a new transformer + a diodes bridge + a cap (to filter the 100Hz sine ondulations). The LT3045 card will regulate to 3,3V as expected. The USB ground and the I2S ground are no longer linked (like in previous option). But in some manner, I’m wondering if the fact to use a transformer will not bring some noise to the DAC (don’t forget the DAC’s I2S ground input is connected at the end). On this point it’s not clear in my mind, as I said I’m not a god in electronic, especially about EMI and so on, sorry.
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Use a battery (6V, 9V or even 12V) to power the LT3045 3,3V regulation board that will power the I2S/LVDS HDMI module. The battery’s ground is, as well, connected to the isolated 3,3V of the Hecate. By this way, there’s no ground loop possible, the ground takes end at he Hecate’s I2S output level. This sounds a good option, the battery is not heavily loaded (only 30mA) but it is not the most convenient, I mus admit.
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Use 2 banks of SuperCaps. The first is “energized” by a PSU, while the second is used as PSU to feed the LT3045 3,3V and then the I2S/LVDS HDMI module. You need a load-balancing mechanism to switch from 1 bank and then to the other bank when this last is discharged. This option seems the best, and should be simpler to implement than a LPS1 UltraCaps from Uptone Audio. The 3,3V/30mA we need should not require big SuperCaps and working at 3,3V is an advantage. It is easy to find affordable SuperCaps at 5V, while higher voltages are more difficult to find or setup. Using SuperCaps instead of a battery (option 3) is a proven advantage in audio if you look at the feedbacks on the LPS1 UltraCaps. Many UltraCaps have a very low internal resistance compared to the internal resistance of a battery.
The NanoPat is, for the moment, built on the Option 2 design, although my initial aim was to build the Option 4
. I’ve put a low ESR audiophile capacitor 3900uF 16V from Panasonic (the one you noticed on the picture) in order to filter the last stage. I believe the LT3045 has its own filtering cap, so I don’t know if the Panasonic brings something or not. In my open questions in a previous post, I still have the interrogation about the importance to power the Hecate’s input (as well as the SBC that feed the Hecate with data) with as much care as we take for the HDMI module…
You didn’t talked about the SBC or computer you use and the USB cable between it and the Hecate board. How do you separate the 5V of the computer to the 5V to feed the Hecate ? 
EDIT: NanoPat is built on Option 2 (not 3)