As you say, this house rule works well for your group. A new player objecting to a house rule they don't understand is no reason for you to change it.
And they don't understand the point of the house rule. They've observed their fellow players and seen the rule's results, and are trying to adjust their actions to fit what the rule is meant to encourage — and they've misunderstood. Instead of engaging in new, creative roleplaying, they're just copying what they see other players do in a kind of cargo cult roleplaying.
This is also the source of their frustration: they think they've solved the puzzle, and they perceive you denying them the reward for solving it. They haven't understood it though, and rewarding them for missing the point will not help them integrate into your group.
Having put in effort and missed the "target" of understanding the point of the bonus experience, though, you should meet them halfway:
"It's clever how you noticed what the other players did to earn the bonus experience. That's great, but it's only halfway to getting the point of the bonus. The bonus is a reward for individual creativity in roleplaying, so simple copying doesn't earn the bonus.
"Does that make sense? This is an important tool that has worked for us, and I want you to understand what it's for so that it works for you too."
You need to emphasise that you're willing to help the new player get up to speed with the group. Emphasise that, because that's what the conversation should be about — not about whether the rule is useful or not (you already know it is for your group), and not whether the player gets to challenge the award (that's not their job in the group dynamic). Just take it as given that this is how it works, that's not up for discussion and don't bring it up for discussion. Move past those givens without even mentioning them, and instead move right to discussing with the player the point of it and how they can reach that point.
P.S. — Contrary to recent online RPG community fashions that resist XP rewards, this is a common and widespread way for RPGs and groups to work, and entirely functional when done fairly and impartially. Current fashion is simply that — passing fashion. Fashion is not legit grounds to tell you it's a bad way to run your game.
A creature typically can't determine another creature's remaining hit points
However, a creature can come close using spells like deathwatch and detect animals and plants, and optional rules like Wound Thresholds will allow more accurate guesses as to a creature's hp, but, overall, a creature's current hp is usually information possessed only by the GM.
Since Pathfinder monsters typically have a predictable number of hp based on CR, a successful Knowledge skill check may give a canny player a means to guess at a particular monster's hit points and, further, allow the PC to glean as "a piece of useful information" how many Hit Dice the creature possesses, if the GM's comfortable revealing that as one of the creature's "special powers or vulnerabilities."
The advantage of the power word spells is that they are ranged spells with no saving throws, a virtual saving throw being provided by the creature's remaining hp. And, while I know the power word spells' disadvantages are many, a caster that knows one or more of such spells and also knows a monster's remaining hp eliminates any guesswork in a way that's superior to the typical caster knowing a monster's current actual Fort, Ref, and Will saving throw bonuses! If a hp determination technique's available, a power word spell's caster no longer need guess and can word foes with impunity. Hence, unless a special ability or spell grants such an ability, I'd advise against a house rule permitting a creature to determine another creature's hp.
Best Answer
This list from Rogue Eidolon's Guide to Fighters should come in handy. The guide has them rated, and it's always worth having a proper build in mind so you know what to ask your DM for. But if you need a generic "wishlist of stuff that a master archer fighter would want" take a look below.
Armour:
Weapons, When it comes to a Fighter, the weapon makes the warrior. Thus, weapons are in their appropriate sections for each build.
Rings:
Wondrous: