It's impossible for Lucky to apply to a passive check, because a passive check doesn't involve a roll on the part of the character. They are treated as having rolled a 10.
It's also worth noting that Survival checks for navigation are generally taken against a set DC - the DM decides how difficult the task of navigating the current set of terrain is, and the character rolls to beat that DC. Using characters' passive skills does not change that. If you're using characters' passive skills for navigation, then if their passive Survival score is equal or higher than your set DC they'll succeed every time, and if it's lower they'll fail every time.
If you change things up by using the characters' passive checks and making an opposed check against them, you're effectively treating the terrain as though it were a living opponent trying to outmatch them - much as would be the case with a creature using Stealth to hide from them.
Yes, this is one of the misgivings I have about using passive Survival; it's either too static or I'm treating the terrain as if it's a creature. I've seen some DMs suggest rolling for traps versus passive Perception, and I've never much liked that idea. The difference here is that while each trap is unique with its own DC and once it's encountered the next one may be different, terrain types have the set DCs you mention. I think this makes passive checks even more of a problem, unless the DC is converted into a roll.
The downside of this is, as you've correctly surmised, that you've given them the disadvantages of making random opposed rolls with none of the abilities they might have to mitigate those disadvantages. Under the circumstances, it would not be unreasonable to make some allowances - i.e. allowing a player who's earned a point of Inspiration to spend it to add +5 to their passive score for the check. Equally, you could apply the halfling's Lucky ability by choosing to re-roll your opposed terrain check if you roll a natural 20.
I think spending a point of Inspiration to add advantage to your passive score is within the rules, considering that a passive check is a type of ability check. I wouldn't feel obligated to apply Lucky, however, since the halfling is already getting the equivalent of rolling a 10, and Lucky only triggers on a 1.
I'll assume here that what the player wants to accomplish is getting across the Old Forest (if I wasn't sure I would always just ask - I'd probably ask them why they said "stay together", I'd ask if they actually walk at some significant distance to each other - for now I assume they don't). Also I assume that there has been some discussion before and the player agreed on this strategy (because I usually don't let a single player declare "we" actions, in fact I'd ask every single player what he's doing while traveling, I'd also ask Strider how he wants to accomplish navigating through the Old Forest).
Going by those assumptions, I'd handle it similar to how I handle stealth checks: There's an immediate consequence after the roll.
So:
Player: "We're heading south-east across the Old Forest. Strider is leading the way to make sure we stay together and don't get lost."
DM: (decides for DC required to succeed, e.g. 13) Ok Strider, roll survival.
Strider: 9.
DM: "You start traveling through the forest with Strider leading the way. He has some trouble finding the right path. You eventually see an opening and head straight to it, but it seems this isn't the end of the forest, but rather a monster lair! You're getting attacked. Roll initiative."
That's a fine consequence, although if I was going to use something like that I might foreshadow that running into a monster's lair is one of the risks associated with navigating through this particular forest.
To me, the more obvious consequence is that they do not succeed in going south-east, but instead wind up going in some other direction due to the difficulty of finding their bearings in the dense (possibly enchanted) forest. An encounter with monsters may be an indirect consequence of the extra time it may take them (and thus the increased chance of random encounters) to get back on course should they decide to do that.
Edit: I might ask you back, why do you think the situation where the players are lost but don't know they are is of any benefit to the adventure here? What would you do? They will certainly notice they aren't out of the forest, even though that's what they wanted to accomplish. Or do you tell them they made it through the forest even though they haven't? Certainly not. What else? Make them stuck inside the forest forever? Or let them roll survival until they succeed?
Well, they wanted to go south-east with Strider acting as navigator. That's their action declaration for this "exploration turn". The basic pattern is something like this:
DM: As you break camp this morning, you're between the Road, which you can see running from east to west about a mile to the north of you, and the Old Forest, the first eaves of which form a line to the south of you, parallel to the Road. What are you going to do today?
Player 1: It's going to take us too long to go around the forest to get to the Barrow Downs, so I think we should cut across the forest by going south-east.
DM: Is everyone in agreement with that decision?
<group confers>
Player 2: Yes, we're heading south-east across the Old Forest. Strider is leading the way to make sure we stay together and don't get lost.
DM: Okay, Strider, what's your passive Survival score? I'll be checking that against the terrain to make sure you stay on course. I'll also need to know your travel pace and marching order.
Player 2: It's 15. We'll go at a fast pace; we need to get to the Barrow Downs by nightfall. Everyone, follow me!
<group sets marching order>
DM: You head into the forest. It's very dense and slow-going. By mid-day you've covered a distance of only 7 1/2 miles, even at a fast pace.
<rolls navigation check resulting in failure; determines party has gone south instead of south-east>
DM: You're surrounded by trees in every direction. What do you do?
Player 1: If we stay the course, we should be out of the forest by dark.
Player 2: Right, let's keep going, keeping our south-east bearing.
DM: Okay, so you continue on through the forest.
<rolls navigation check resulting in success>
DM: As the light of day begins to dim, you still haven't reached the edge of the forest.
Player 1: Oh no! We must have gone off course somewhere!
DM: You come across a low hill, however, and from the top you can make out the dark line of the Barrow Downs in the distance, about 5 miles due east.
Player 3: Oh well, there's no way we'll make it there before dark. We'll just have to set up camp and get a fresh start on it tomorrow!
So with a failure, the party goes off course in a random direction, while with a success, they realize they were lost and have a chance to course-correct.
