Before changing your setup which appears to work with the PS4, I think you need to test your headset wire and chat adapter. Try joining a chat and connecting your headset directly into your chat adapter and see if your friends are still getting an echo. You should turn on the party chat overlay or leave the chat window open so you can see when your mic is giving feedback.
Assuming you don't have any issues with your adapter or headphone wire, we can now try different setups.
This would be your current Mayflower setup if you are following the instructions Alan provided from the manual.

However, if you are sending party chat through the Toslink cable (blue) your setup would be a little different (remove green lines/adapters to the chat adapter). I'm pretty sure the green section is where Mayflower got their Xbox instructions wrong. If you did connect it in this manner, you are now feeding an amplified signal directly in-line with the unamplified sound output from your chat adapter. I believe this would lead to lots of feedback and quiet party chat. If you remove this entirely and change your chat output to optical, your sound should be perfect.
Now lets look at the mic connection. I believe the problem is your headset is a 4 pole and your splitter converts this to two 3 pole connections which share the same ground. This wouldn't be an issue with PS4 because all signals are sharing the same ground from the Mayflower. However on Xbox, when using the mic passthru you are still dealing with two separate power supplies:
1) Mic powered from controller adapter
2) Speakers powered from Mayflower.
You would think the mic input would change the ground, but I believe it simply just forwards everything back to your chat adapter. I may be completely wrong, but other complaints from Xbox users lead me to believe this is the problem.
This brings me to the addition of the Modmic. This is a sad compromise because now you'd be adding a second mic onto your headset so you no longer share ground between mic and speakers. I hope you find a way to avoid this, but I suspect this new arrangement is the only way to prevent ground issues and your mic from echoing.

It's also possible that the mic and speaker inputs inside the ARC share the same ground. If so the feedback would be caused by the internal grounding. If that's the case, adding a Modmic wont help, we would instead need to connect the mic directly to the controller (red dashed line).
I apologize for such a lengthy post and unfortunately I don't have the Arc so I can't test and give you a solution.