Is there any conjecture that has been proved to be solvable/provable but whose direct solution/proof is not yet known?

It is known that there is an even integer $n\le246$ such that there are infinitely many primes $p$ such that the next prime is $p+n$, but there is no specific $n$ which has been proved to work (although everyone believes that every even $n\ge2$ actually works).

I am not sure this answers your question but it seems to come close. The Wikipedia article Skewes number states

In number theory, Skewes's number is any of several extremely large numbers used by the South African mathematician Stanley Skewes as upper bounds for the smallest natural number $x$ for which $\, \pi(x) > \textrm{li}(x) \,$ where $\pi$ is the prime-counting function and $\, \textrm{li} \,$ is the logarithmic integral function.

It further states that

All numerical evidence then available seemed to suggest that $\, \pi(x) \,$ was always less than $\, \textrm{li}(x). \,$ Littlewood's proof did not, however, exhibit a concrete such number $x$.

The problem is that even though the existence of the number $x$ has been proven, we only know huge upper bounds on the first such number.

Perhaps a better example is from the Wikipedia article Ramsey theory where

Problems in Ramsey theory typically ask a question of the form: "how many elements of some structure must there be to guarantee that a particular property will hold?"

These Ramsey numbers have the property that

Results in Ramsey theory typically have two primary characteristics. Firstly, they are non-constructive: they may show that some structure exists, but they give no process for finding this structure (other than brute-force search). For instance, the pigeonhole principle is of this form. Secondly, while Ramsey theory results do say that sufficiently large objects must necessarily contain a given structure, often the proof of these results requires these objects to be enormously large – bounds that grow exponentially, or even as fast as the Ackermann function are not uncommon.

Thus, in theory, some of these enumerative numbers exist but they are too huge to write explicitly. In other cases, there exist small bounds but it is very hard to narrow the bounds.

It's known that no matter the size of the board, the first player has a winning strategy in Chomp, but an explicit winning strategy is only known for smallish boards.

Well, this might be an example of what you are asking for:

Take the question "are there irrational numbers $a,b$ such that $a^b$ is rational?"

A quick way to see that there are is to consider $\alpha =\sqrt 2^ {\sqrt 2}$.

Either $\alpha$ is rational or irrational. If it is rational, then we are done. If it is irrational then consider $\alpha^{\sqrt 2}=2$. Either way, we can find an example.

To be sure, it is possible (though quite difficult) to show that $\alpha$ is irrational and there are other ways to get direct examples (such as $e^{\ln 2}$) that settle the problem . But this indirect proof is so simple it is, I think, worth study.

Define the following number:

$$K = \begin{cases} 1, & \text{if Riemann Hypothesis is true} \\ 0, & \text{if it is false} \end{cases}$$

Since the Riemann Hypothesis is either true or false, the above constant $K$ is well-defined, mathematically. It is one specific number. But no one knows whether it is 0 or 1 (because knowing it means knowing the answer to the hypothesis).

I know this example is very artificial, but nevertheless I thought it was interesting to share - basically any unsolved problem in mathematics can be converted in an object that certainly exists but knowing what it is becomes equivalent to solving the problem itself in the first place.