Specific ways/practices to help die with equanimity, (less "self") and perhaps even awakening at death-time?

Equanimity while not having penetrated, abound desire for sensuality, will bring one anywhere, and sure not in refined states. And lost any sense of self, while still consuming, if good in the realm of "mindless-beings". So good to develop proper conceit in seeing the dangers in the world (sense) and seek out for people teaching the Dhamma not training consuming fools and suggestions that they could escape the debts. Only debtless one is able to progress, having abond desire for foremost gross food.

Haven't faced death yet, if faced then don't remember if that was near death experience or just faintness.

But as heard from others about death, it causes extreme level of pain at physical level, all life passes in form of picture in mind, things seem to start breaking up and decisions are made according to the behaviour pattern of subconscious state.

Let's analyse these:-
The basic equanimity consists of these things :- 5 precepts, right thinking / thoughts, 8 worldly concerns, fear.

So basic equanimity should simply means to maintain a fearless right thinking or right thought process or right.. 8 folds by remaining alert against 8 worldly concerns using 5 precepts as tool and seeing end result as one of the 4 noble truth.

During extreme pain, it should be highly impossible to maintain "right thinking in form of 5 precepts" without meditation practice. Even when strong swaying away of emotions is there then mind pictures can be aligned to result into right process only when one is having a practice previously.

Not to run away, not to make wrong decisions, not to flow with defilements etc. should be the points to keep during that time. And apart from all, observer space or an observer like third perspective should be there.

So it's all upto you, what you wish to practice.. all, physical only, mental only, dhamma in form of fabrications only.

Note: for the first time today, I got introduced(or learnt) by the basic definition of equanimity while writing... Before it was all scattered as if what should I call equanimity? So untill enlightenment, as everything is just 99% correct and 1% space for rectification hence this might also not be 100% correct.
(Hindi version could be there for better understanding/feel in local language).

How to practice at the moment of death:

Buddhadasa World as Empty - Buddha's Teachings on Voideness Remainderless Quenching of the "I": Extinguishing with nothing left

There is nothing in this or any world that is worth to have or to be - the mind should have no hope of having or being anything at all. Truly, nothing is worth having or being, at any time or in any place. The mind tends into its own dissolution. With no desire to have or be anything, the mind dissolves into suññata.

When the time has really come for the mind to cease, revive the feeling that nothing anywhere is worth having or being. If that feeling is present in the mind at the moment it ceases, it will reach nibbana inevitably. Have the body and mind cease with the feeling that nothing is worth having or being, then it will realize nibbana in that physical death itself.

When the time of death has trully come, let this feeling be present. Remember that when close to death the mind will gradually slip away. As the body runs down near its end, awareness will gradually disappear. You will forget more and more until you forget everything. But the awareness that nothing is worth having or being can stay as the mind's companion to the very end. Keep the feeling of the remainderless quenching until the very end - nothing is worth having or being. Realize suññata in the last second of life.

Suchart If you do not want rebirth, you will have to stop your mind from dreaming, from going to a new life. That's what the Buddha did. He stopped his mind by cutting off the desire in his mind. He had no desire, and when there is no desire, the mind doesn't generate anything, doesn't generate any thoughts, and doesn’t have any force to drive it to go into a new body. So the key idea here is to let go of everything. Nothing belongs to you. If you cling to it, you will suffer. If you let go, you will be at ease and be peaceful.

Dune Atulo When time comes to die, make the mind one, then stop focusing and let go of everything.

You can also check MN143 - Anāthapiṇḍikovāda Sutta https://suttacentral.net/mn143/en/suddhaso

Verse 6 from the Dhammapada:

There are those who do not realize that one day we all must die. But those who do realize this settle their quarrels.

If there were any quarrels it might mean a lot, to have ended them.

1.please learn about four brahma viharas and specially about Upekkha (equanimity as you have mentioned) and practice it constantly and continuously.

1. and try to deepen your VIPASSANA understanding, because that's the main way of getting the true equanimity! Suppose a practitioner even without practicing equanimity, he can become enlighten by just practicing VIPASSANA well.in that case as an arahant will automatically become equanimity.