Is it possible to have awareness without a self-sense? Is it possible to have awareness without a self-sense? Does an awareness presuppose  a self being aware no matter how refined or subtle that sense of self might be? Or to reframe the question slightly - can awareness be non-dual i.e. the being aware of the subject without the object.

Many thanks as always
OP: Is it possible to have awareness without a self-sense?

Yes, why not? It's very hard to imagine how enlightened ones think about what they use, their parents, relatives, etc... without self-sense. They use things with knowing that this is not me, this does not belong to me, this cannot be kept as I wish. How beautiful that thinking is!
Do you know even a non-returner lived in a home, did a job (without earning money), and looked after his old blind parents? I quoted here Kassapa Lord Buddha's own words about him, to understand how he did those without self-sense in order to imagine a little about how enlightened ones think.

But Ghaṭīkāra doesn’t get upset, nor will he. Ghaṭīkāra has gone for refuge to the Buddha, the teaching, and the Saṅgha. He doesn’t kill living creatures, steal, commit sexual misconduct, lie, or take alcoholic drinks that cause negligence. He has experiential confidence in the Buddha, the teaching, and the Saṅgha, and has the ethics loved by the noble ones. He is free of doubt regarding suffering, its origin, its cessation, and the practice that leads to its cessation. He eats in one part of the day; he’s celibate, ethical, and of good character. He has set aside gems and gold, and rejected gold and money. He’s put down the shovel and doesn’t dig the earth with his own hands. He takes what has crumbled off by a riverbank or been dug up by mice, and brings it back in a carrier. When he has made a pot, he says: “Anyone may leave bagged sesame, mung beans, or chick peas here and take what they wish.” He looks after his blind old parents. And since he has ended the five lower fetters, Ghaṭīkāra will be reborn spontaneously and will become extinguished there, not liable to return from that world.

~ Cited from: Ghaṭīkāra Sutta (MN 81)
I have provided this answer by assuming the OP's intended sakkāya-ditthi by the word, self-sense. To my understanding absence of self-sense (sakkāya-ditthi) is not something that totally abandon you from outer world or conventional relationships. We can understand this by reading the following conversation between Ghaṭīkāra and Jotipāla.

Jotipāla: Dear Ghaṭīkāra, you have heard this teaching, so why don’t you go forth from the lay life to homelessness?
Ghaṭīkāra: Don’t you know, dear Jotipāla, that I look after my blind old parents?

Note: This is how I understood. I may be wrong but not Dhamma.