Do you have an audio interface ?
You could measure the latency wich is to expect very low but measurable.
Connect the DAC output to the interface input and play a music from a DAW.
Record this into new tracks twice, once with EQ ON and once with EQ OFF, zoom in and measure the distance between the two recordings, this is the EQ latency.

Unfortunately, I do not have another audio interface, not possible to measure the latency on my own. I wonder, theoretically or technically, about the impact of the internal EQ on overall output latency.

OSU! seems to be an interesting game.
Now I am getting the point, what your requirement is, thanks for the background of your question.
Wouldn't the easiest solution simply be to disable the EQ when playing this game?

If you would had the ADI-2 Pro FS then you could measure it yourself pluggin a loop between Analog output and Analog input and using the Software RTL utility to measure the RTL with PEQ enabled disabled.

I can try to perform this later for you if you provide me your EQ settings (PEQ, dynamic loudness, ?).
I think the type of D/A filter is irrelevant here, you only want to know the difference in terms of EQ functions.

Measuring result

Loudness, EQ, cross-feed, D/A filter do not change the measured RTL which stays constant.
The RTL depends solely on sample rate and ASIO buffersize.

Minimum ASIO Buffersizes at lowest and highest sample rate and the resulting RTL:
-   32 samples @44.1 kHz: 118 samples RTL = 2,676ms
- 128 samples @192  kHz: 358 samples RTL = 1,865ms

I think the reason for this is, that the FPGA (like a CPU in a PC) is so fast, that it can process this all in parallel / in time similar to a CPU in a PC.

If you think about this further, then it would be a hard time for the ASIO driver to tell the host application the latency for latency compensation. Furthermore, for this use case only sample rate and ASIO buffersize have relevance, not every single EQ setting.

When working with a DAW you can notice clearly, that using more and more VST (compressor, EQ) does not result in more latency. It results in more CPU consumption. And you get audio loss, if one or more cores of the PC are not fast enough any more processing audio in time. The transfer of a few audio channels themselves alone does not create a high load for the PC … it's more the processing in time and if you have to transfer many channels. Another example, utilizing a 10Gbit network card fully (which is way more bandwidth) does not even create 1 percent CPU load with the right card and the right driver.

The setup

(*) this USB connection only for management using ADI-2 Remote software)

PC---USB2-----------------------------------------------------+ (*)
|                                                                                 |
+----USB3---UFX III---ADAT1 OUT------->ADAT IN---ADI-2Pro FS R BE---ANOUT-L--------+
                            \                                                    /             \                                    |
                              \                                                 /               +----------ANIN-L---<----+
                               +--ADAT1 IN<------ADAT OUT---+     

Measurement with RTL utility v1.0.7
Windows 10 Pro, 22H2
MADIface driver 0.9831

+1