Interesting challenge. I also have limited time to game, usually just a few hours here an there. But you have to invest time in at least one complete play through to get the story and character stuff. Arguably, you need two complete play throughs, because of the Templar vs. Mages choice you are forced to make. Once you've heard all the story/dialog, you can just skip all the talking in follow-ups.
So my advice would be thus:
1) Play on Casual difficulty. You'll spend less time in combat and less time crafting gear to survive combat. For a first PT, it's a reasonable compromise. You can always up the difficulty later.
2) Complete Wicked Eyes, Wicked Hearts (the "Orlesian Ball" you mentioned), talking to everyone. You can't really cut any corners unless you know what you are doing, and if it is your first time through, you don't. You may even need the Caprice Coins to bail you out if your approval is too low. I'd recommend using one of the many "Find all the Halla Statues" guides and walkthroughs that have been written up, that will save time. You don't need all of them, but not having enough when you need them, or using them for trivial things when you need them for more important things will waste time.
2) If you haven't already, complete Here Lies the Abyss. There's less talking in this, so you can go faster. Just follow the quest markers, don't take any detours to the side -- you'll be giving up some nice stats bonuses, but saving time on skipping optional combat. Make sure you are at least in the middle of the recommended XP range (12-15, I think). It has relatively difficult battles, even on Casual, and reloading because your party gets wiped out wastes time. Doing some crafting wouldn't hurt either, though that uses a lot of time. If you have gold to spare, just buy good stuff from merchants, like the Black Emporium. To level up, do side quests or go kill a dragon or two.
3) At this point, you will either be short of Power or short of XP to do the next main quest stage, What Pride Had Wrought. Easiest way to level up is close rifts in places you've already visited (Hinterlands, Storm Coast, Crestwood, Western Approach, and Forbidden Oasis if you opened it), but if you must open a new zone -- open the Hissing Wastes (Caution: rated Level 19-23, but you can survive it under-leveled). You'll level up fastest there just by closing rifts. Make sure you have a fast mount, because you'll be doing a lot of riding. A good/fast way to exploit the Wastes is to discover the camps on the left side and center of the map. The Mountain Fortress camp on the right side is also good. Having camps to fast travel to will save time. Just clear all the rifts around those camps and you will level up quickly and/or gain enough power to open the next stage. (An alternative to the Hissing Wastes is the Exalted Plains, Level 10-13. It is much harder to get around, but if you just focus on clearing the Ramparts, you'll level up quickly).
4) Do the side quest Before The Dawn if you sided with the Mages, Under Her Skin if you sided with the Templars. This will also help you level up and will shorten the boss battle in the next stage of the main quest.
5) Complete What Pride Had Wrought -- this actually is one of the quickest stages IF you make the decision to "jump down the hole" instead of doing "the rituals". The rituals take forever. However, note that jumping down the hole is an important RP choice and will make it impossible to ally with SPOILERS. It's not a big deal either way, but just wanted to make sure you know. Make sure Morrigan drinks from the Well. This will save time later.
6) In your initial world state, did you set up Morrigan to have a child? If you did, you have to spend some time in a side quest. If you did not, you can skip the side quest.
7) Finish the rest. You should be levelled up enough to run right through to the end from this point on. Having Morrigan drink from the Well will save time fighting Coryfishes pet dragon.
That's about 8 to 16 hours of play, give or take, depending on how much time you spend on crafting or running around doing Companion quests (see below).
If you want to do Companion/Inner Circle quests, I recommend these in descending order of interest (highly subjective, YMMV):
- Sera (although it requires some War Table missions, which take time, although you can start them at the end your game session and they'll be done when you come back the next day).
- Iron Bull
- Cole
- Solas - but only if you have already opened the Exalted Plains
- Varric - the Bianca/Valammar one, not the Red Lyrium one
- Leliana
- Cullen
- Josephine (actually more interesting than previous 2, but very long, involving multiple stages and War Table missions -- even with the short cut)
- (skip the rest, they are all either fetch quests, or go here and kill these dudes quests)