Hey Venerable 
To provide some context for others: We had a PÄli class where we discussed this term, and we ran out of time to discuss the details. I thought Bhante Sujatoās thoughts could clarify things.
I realize now you are mainly asking as a practitioner, which I can share some of my opinions on as a fellow practitionar, but in light of our PÄli class and this topic, Iāll try to also focus on some more theoretical points.
First of, I can relate to the feeling of being a bit lost when you donāt really know what the Buddha is saying. But unfortunately I canāt tell you what the Buddha meant, because thatās the question we all have! 
I can tell you what I think he may have meant, though.
Other people will also give their opinions of what they think the Buddha meant with parimukha (or with anything, really!), but in the end weāll have to decide for yourself, unless weāre happy to just trust somebody else. Or unless weāre happy to stay undecided about some minor details, which is why I explained in our class that the practice of ÄnÄpÄnassati doesnāt hang on this word alone. The actual practices starting with āsatova assassatiā are clear enough, to me anyway.
Either way, we all have to be careful in the way we interpret things. Something being helpful doesnāt make an interpretation technically accurate, for example. Whatās helpful for one person isnāt necessarily helpful for another, so then it becomes a personal interpretation, one that perhaps isnāt the Buddhaās.
Thatās why I (like Bhante Sujato) prefer to investigate what the term parimukha means elsewhere, and donāt get my personal opinion involved if possible. Now, setting aside commentaries, in the PÄli I think there is only one instance of parimukha outside of the phrase parimukhaį¹ satiį¹ upaį¹į¹hapetvÄ, and there it means around the mouth (or perhaps around the face?), in context of shaving in the Vinaya. Itās not much to go by, but since itās all we got, it seems to me that this meaning likely also applies to parimukhaį¹ satiį¹ upaį¹į¹hapetvÄ. It does sort of work with ÄnÄpÄnassati as well, if you take the area of focus on the breath (e.g. nose) to be the area āaround the mouthā.
It also fits the order of things, you sit down, cross your legs, straighten your body, and then you attend to the breath at the nose. If it meant making mindfulness a priority, or some other idiomatic meaning, I donāt know why you would only do that after sitting down and so forth, and not before.
A significant problem for this interpretation of āaround the mouthā, though, is that parimukhaį¹ satiį¹ upaį¹į¹hapetvÄ is in quite a few places used without being followed by ÄnÄpÄnassati, like DN2, AN9.40, MN32, MN107, AN10.99, etc.
Perhaps ÄnÄpÄnassati is still implied to be practiced in those texts, though? Because what follows the phrase doesnāt necessarily exclude that possibility.
Or perhaps the phrase was inserted by accident? Because elsewhere outside of ÄnÄpÄnassati we have a shorter version of the āgone to the wildernessā phrase, like SN41.7: āItās when a mendicant has gone to a wilderness, or to the root of a tree, or to an empty hut, and reflects like this: [ā¦]ā Perhaps the reciters in some suttas accidentally added the rest to this, including parimukhaį¹ satiį¹ upaį¹į¹hapetvÄ, even if the context was not ÄnÄpÄnassati. It would be interesting to see what the parallels to such suttas say. If the parimukhaį¹ satiį¹ upaį¹į¹hapetvÄ is missing there, that answers it for me. Then it means parimukhaį¹ does indeed have to do specifically with ÄnÄpÄnassati.
Another option given by some (sub?)commentary is that parimukha means āwhatās in frontā in the sense of the meditation object that you have āin frontā of you. .
Well, the verb upaį¹į¹hÄti together with sati essentially mean āto be mindfulā. So, yes, regardless of what parimukhaį¹ means, satiį¹ upaį¹į¹hapetvÄ tells us that mindfulness is already present. One is already mindful. Thatās why I donāt think parimukhaį¹ means that you make it a priority. You would have done that already before āhaving made mindfulness presentā (satiį¹ upaį¹į¹hapetvÄ).
I also think ÄnÄpÄnassati (or any kind of sitting meditation) requires that we already have a certain amount of awareness. Consider for example MN107:
When they have mindfulness and situational awareness, the Realized One guides them further: āCome, mendicant, frequent a secluded lodgingāa wilderness, the root of a tree, a hill, a ravine, a mountain cave, a charnel ground, a forest, the open air, a heap of straw.ā And they do so. After the meal, they return from almsround, sit down cross-legged, set their body straight, and establish mindfulness in front of them.
Although also here this isnāt followed by ÄnÄpÄnassati, the idea is the same, that you have some kind of mindfulness already before you start sitting meditation. So also here, you already made mindfulness a priority before you sit down.
Sukha and pÄ«ti are said to arise only at āstepā 5 of the practice, so Iād say you donāt have to start with them. Itāll help, though, if you can bring them up earlier.
ÄnÄpÄnassati is in AN9.3 also said to be useful for abandoning thinking, so you clearly donāt have to be super quiet and at ease before doing it.
Well, there is āa noble disciple, relying on letting go, gains immersion, gains unification of mind.ā (e.g. SN48.10) Thatās the whole instruction we get in some places! 
And it is followed by the jhÄnas. ÄnÄpÄnassati is one way to get to the jhÄnas, but not the only one. If itās not working for you, then do something else!
When I taught ÄnÄpÄnassati, I made a point to emphasize this part of the sutta:
There are such mendicants in this Saį¹ gha. In this Saį¹ gha there are mendicants who are committed to developing the meditation on love ā¦ compassion ā¦ rejoicing ā¦ equanimity ā¦ ugliness ā¦ impermanence. There are such mendicants in this Saį¹ gha. In this Saį¹ gha there are mendicants who are committed to developing the meditation on mindfulness of breathing.
Where does it say that mindfulness of breathing is better or more preferrable or that we should do it? Nowhere. We can do all these things (and more) and theyāre just as worthwhile.
Btw, equanimity is also basically a practice of letting things be, in my eyes.
Also, this is straying from the texts a bit, but as long as we understand what the hindrances are and how to overcome them, we donāt have to stay rigidly within the practices mentioned in the suttas. We can invent our own. I do āwhateverā meditation. I just sit there and say to myself āwhatever!ā Whatever happens, itās all fine. It removes lots of desire and restlessness.
Likewise with parimukha. I may not know with certainty what it means, but āwhateverā, I can still meditate!
See SN47.46, SN47.47, AN6.117, AN10.61, remembering that ÄnÄpÄnassati encompasses the four satipatthÄnas (āthe four kinds of mindfulness meditationā in V. Sujatoās translations).
I do hope all that helps alleviate some doubts.
And although we got a bit sidetracked with parimukhaį¹, I also hope you understood the basic grammar we talked about 
agree 
a lot.
, adverbs donāt have cases. See for example Digital Pali Dictiory, which glosses it as āindeclineableā, meaning it has no case. It does have an accusative form but it is not āin the accusativeā I would say. A bit of a technicality.