C++ priority_queue: how to modify STL template to find & delete based on pair’s 1st element?

I want to modify the <queue> STL so that it can find any element from the priority_queue, then delete it from the pq. The code is based on the answer by alexm from How to remove element not at top from priority_queue?

However, what I want to achieve is more complicated: the element in the priority_queue is pair<char, double>, and I want to achieve a min-heap. Thus, I modified the code as below:

#include <iostream>
#include <queue>
using namespace std;
using item_type = pair<char, double>;

template<typename T>
class custom_priority_queue : public priority_queue<T, vector<T>, greater<>> {
public:
    bool eleExist(const T& value) {
        auto it = lower_bound(this->c.begin(), this->c.end(), value);
        if (it != this->c.end())
            return true;
        else
            return false;
    }
    
    bool remove(const T& value) {
        auto it = lower_bound(this->c.begin(), this->c.end(), value);
        if (it != this->c.end()) {
            this->c.erase(it);
            make_heap(this->c.begin(), this->c.end(), this->comp);
            return true;
        }
        else {
            return false;
        }
    }
};

int main() {
    custom_priority_queue<item_type> pq;

    // use list to fill priority_queue later
    // larger number means higher priority
    initializer_list<item_type> il {
        {'A', 10},
        {'B', 12},
        {'H', 12}
    };
    // iterator through list to fill into priority_queue
    for (const auto& item : il) {
        pq.emplace(item);
    }
    
    if (pq.eleExist({'A', 10})) {
        cout << "A exists. Thus remove it" << endl;
        pq.remove({'A', 10});
    }
    else
        cout << "A doens't exist" << endl;

    // print priority_queue
    cout << "Currently in priority queue:" << endl;
    while (!pq.empty()) {
        cout << pq.top().first << ": " << pq.top().second << endl;
        pq.pop();
    }
    cout << endl;

    return 0;
}

Unfortunately, I can only use eleExist and remove functions with the full pair in my main function. What I really want to achieve is, to find element in priority_queue based on the 1st element in pair, and is able to delete the pair element based on the eleExist function. How do I modify my derived class custom_priority_queue to achieve that?

Thanks.

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