Don't punish your players for doing well.
A player having a +9 to Perception at an early level really isn't a problem. Sure, starting at level 1 with a 20 in Wisdom and Expertise in Perception is a little minmax-y, but that isn't an issue. The player obviously geared the character towards it, and punishing them by reducing their stats would be incredibly unfair to them, and punishing the rest of the group by making perception checks that are impossible for them unless one player gets lucky is just as unfair.
So, what can you do?
Don't rely on failing Perception checks to make an interesting dungeon.
The secret door? Now it's a magical door. They don't need to roll to see it; instead, they need to determine how to open it.
A tripwire trap? Nope; now it's a giant purple crystal that explodes if the players make too much noise. Instead of rolling to see it, now they have to figure out a way to get the party around it without dying.
An ambush of enemies? Now it's a group of enemies so large or fierce that the party decides they need to avoid the group rather than spot them and kill them.
You can even keep your usual traps, and instead add things in to make them something that requires an approach other than Perception. Keep the tripwire trap, but now the room is covered in magical darkness so they have to dispel that first. Keep the ambush, but now it takes place in an area with heavy fog, so Perception is at disadvantage.
That said, you shouldn't take away success from this player. It would be very un-fun, for both this player and the party, if you took away all Perception checks. She should have situations in which there is an invisible item to spot, or a trap that can be seen. Players have fun when they get to feel cool, and if they've geared themselves towards a large bonus in a skill, let them use it and give them that cool feeling.
Disclaimer: Possible spoilers about LMoP.
So, I have read through the rewards of the third chapter, plus the "starting point" of the two first chapters. The needed encounters to reach the recommended 4th level can be answered by math. From a point of view of experience, for 4 players, you need a total of \$ 4 \cdot 2700 = 10800 \$ XP for the party.
The first chapter gives you a total of \$1400\$ XP (350/character).
The second chapter gives you a total of \$+2800\$ XP (+700/character).
The Cragmaw castle, from the third chapter, gives you a total of \$+4400\$ XP, assuming
the party clears the castle completely, rescues Gundren and you include the optional Returning War Band encounter.
From the main story, we have a total of \$ 8600\$ XP. That means the Side Quests should give the party at least \$10800 - 8600 = 2200\$ XP. I have split the side quests in four: Agatha (sent by Sister Graele), Wyvern Tor (sent by the Mayor), Old Owl Well (sent by Daran) and Thundertree (sent by Quelline Alderleaf).
\begin{array} {|r|r|}
\hline
\textbf{Side Quest} & \textbf{Total Awarded XP} \\
\hline
\textrm{Agatha} &200 \\ \hline
\textrm{Wyvern Tor} & 1250 \\ \hline
\textrm{Old Owl Well} & 1000 \\ \hline
\textrm{Thundertree} & 3750 \\ \hline
~~\textrm{Westernmost Cottage} & 50\\
~~\textrm{Blighted Cottage} & 50\\
~~\textrm{Brown Horse} & 200 \\
~~\textrm{Blighted Farmhouse} & 200 \\
~~\textrm{Ruined Store} & 400 \\
~~\textrm{Dragon's Tower} & 2000\\
~~\textrm{Old Smith} & 100 \\
~~\textrm{Herbalist's Shop} & 200 \\
~~\textrm{Old Garrison} & 250 \\
~~\textrm{Weaver's Cottage} & 150 \\
~~\textrm{Dragon's Cultists} & 150 \\ \hline
\end{array}
Essentially, the side quest giving the most XP is, by far, the Thundertree, due to the Dragon encounter. It is also the encounter that gives the most valuable magic items - +1 Weapon, 2 scrolls and 2 potions. Old Owl Well also gives a Ring of Protection, which is important.
From that, simply don't take Thundertree from the party, which is also the safest way to learn about the Castle's Location through Reidoth. That doesn't even account for the random encounters.
As a second option, the Cragmaw Castle can be skipped, actually, as long as the party can get \$400\$ XP from the random encounters, which shouldn't be hard.
As a side note, you shouldn't be worried about the PCs being able to find the Castle. The Goblins and Hobgoblins from random encounters are able to provide that info, so the party only needs to wander around for some time.
Best Answer
The Gumshoe RPG was designed to address this issue. The way it does this is by shifting the focus from finding clues to interpreting clues. In a nutshell it assumes when it comes to finding a clue, if the players have the skill they will find the clue. What to do with the clue is a different matter altogether.
Fortunately this concept can translate over easily to D&D 5e. The way I handle situations like yours is that absent an adverse consequence for failure, time constraints or some other circumstance that make the result uncertain, assume the PC succeed given enough time.
Now how does this help you with the Mad Mansion? Looking over the module, the author has a lot of interesting things going on, however, he has a lot of unnecessary checks as well. If the players are unlucky outside of combat even if they are doing the right things then they will be unhappy with the module.
For example in room F3 Study there is this
My view is that unless the character rolls a 1 for sheer bad luck, that journal should be found provided that the player thinks of searching the drawer in the first place. And that is the key to the Gumshoe method.
In Gumshoe a character with Forensics has to say that he is sweeping the room for fingerprints before he obtain any clues about fingerprints. However the problem with this is that the session can easily turn into a game of twenty question which can be unsatisfying in its own way. The way Gumshoe avoids this is by giving a lot of advice about how to construct a train of clues that offer alternative paths to solving the main mystery.
So to make the Mad Mansion more interesting, I would go through the module and make a list of everything that is important to find, and seed the rest of the module with at least two different clues to where a individual item could be found. As the players continue their search build them a picture of how Gerardus lived and where he kept stuff. The good news is that the skeleton for doing this is already in the module. Your focus should be seeding the other rooms with clues. Remember, you want to avoid having an item that can only be found by saying the one right thing in the right room.