It seems best to me, from the information you give, to somehow describe the situation in the way you did here.
To be a bit more precise what I mean. You could do something like this.
Theorem: We have $f(n) = O(c^n)$ where $c$ is given as the solution of 'defining equation for $c$'.
Followed by something like:
Note that in particular the above result implies that $f(n)=O(d^n)$ with $d=1.2345$ as, solving the above equation numerically, one can see $c<1.2345$.
One could also do the dual, i.e., give the numerical value in the result and give the explanation how it arises afterwards. However, I would not just mention the 'defining equation' in the proof but close to the fomulation of the result.
Since you are worried about this: it is not that uncommon to have theorems where the constant arises as a numerical approximation to someting 'complicated.'
In case the defining equation is very complicated (and you are worried about too much clutter in the formulation of the theorem) and/or there is something to be discussed regarding existence of solutions or alike, I suggest to define the $c$ outside the formulation of the result.
Now, I said above, I would not hide the defining equation in the proof. However, I would make an exception to this general advice in case it is really complicated or only understandable in the context of the proof. Still, I then would write a sentence pointing to the place in the proof where it arises right after (or possibly even in) the formulation of the theorem.
In any case, what seems important to me is to make clear for the reader how the numerical constant arises.
One additional aspect to keep in mind is the following question: Will, and if so in which form, readers care about the numerical value?
If the point of the paper is that the constant is better than in early work, of course, the value and this fact needs to be more visible, than if this is the first bound of this form and the value of the constant is by and large irrelevant.