I mean Kurama is in Naruto so wouldn't it be pretty awkward for the former to be present in all of the latter's intimate moments, be them romantic or physiological in nature?
Yes Kurama knows and sees the same things as Naruto so this is included. As he is an old being he has probably seen his own fair share of naked ones and those making love, meaning this is something normal to do for human beings in his eyes.
It being awkward? No I don't think so. For Naruto they have been together since his birth 24/7. They can be considered as one essentially (of course they aren't, but having one without the other would make both someone different). Because Kurama knows the ins and outs of Naruto it is something that isn't awkward for him. For Naruto it would be weird to discus it with Kurama, but not awkward as he is aware that Kurama knows and sees all that he does to since his entire life.
Going by Kurama's personality I don't think he even cares enough to consider these moments awkward.