Like they say on reddit, the 3 doors are not in the suttas. The 3 doors are some inventions from the commentators of some mix of commentaries, abidhammas, and suttas.
Those are not the dhamma. Even the mix of commentaries, abidhammas, and suttas on which they comment is not the dhamma. Even the suttas alone can be blindly said to be the dhamma.
So first, the usual way use of doors is the ''guarding of the sense doors''.
For the signless, there is plenty of that in the suttas, but there is not much about them, just like for ''desire for non-existence''. They are just listed and zero explanation. This is great for commentators who can tack their various speculations in their teachings, but at the end of the day, nobody knows what those means.
So for the signless like here https://www.accesstoinsight.org/tipitaka/kn/snp/snp.2.11.irel.html
the note says
The Signless (animitta) is one of the three Deliverances (vimokkha) by which beings are liberated from the world. The other two
are Desirelessness (appanihita) and Emptiness (sunnata). The Signless
is connected with the idea of impermanence of all conditioned things
(cf. Visuddhi Magga, XXI 67f).
This is the usual interpretation of those weird signless, desirelessness and emptiness liberations. You can read the Visuddhi Magga to read more on this, but the Visuddhi Magga is not the dhamma, only a some commentary by some people.
The usual connection with the fetters is with the births, ie Breaking the first 5 fetters frees you from rebirth in the sense-world, and breaking the remaining 5 higher fetters, frees you from being reborn in the form world and the formless world.
So the non-returners have broken the 5 lower fetters and if they fail to reach nibanna as humans, they get to be reborn as some non-returner devas. Now the difference between those nonfully enlightened devas and the normal devas is here https://suttacentral.net/an4.123/en/sujato
that the normal devas delight in whatever world they live, then they are reborn in hell sooner or later (like any non enlightened people), and the enlightened devas just die like the human arahants.
So to link anicca dukkha and anatta to the fetters, you link them to the realms of births and you can say that the five higher fetters are failure to see anicca, dukkha, anatta of those higher realms , plus the failure to see that what is anicca about those realm is also dukkha, and to fail to see that what is dukkha about those realms is also anatta [ie to to crave for them].