I have 2 comments:
1) Empirically I have a guess for the positive real roots:
Let a scaling-factor $c=\frac1{\ln 2}-1 \approx 0.442$ then
$$ \rho_k \approx (k+3) \cdot c $$
or $$ k \lt {\rho_k \over c }-3 \lt k+1 \tag {for $k \ge$ 5}$$
I checked this up to $k=5000$ so far using the bernoulli-polynomials and internal float precision of 400 digits in Pari/GP (see table below). The idea behind this is to show that either the roots are irrational or that the approximation of the real root to an integer is too bad, which is supported by the small magnitude of the $\epsilon$ compared with the systematic approximation of $k \cdot c +1.5 $ with increasing $k$ to an integer.
2) looking at the equation modulo $k$ it seems, the lhs of the equation is always equivalent to zero, so solutions can only exist, if also the rhs is zero modulo $k$ - here I assume only odd $k$ so far.
Let $w$ be the squarefree kernel of primefactors (all are odd) of $k$. Then the rhs is equivalent zero only if $x$ is a multiple of $w$ and thus only such $x$ can solve the equality. So if we have some $k=p_1^{b_1} p_2^{b_3} p_3^{b_3} $ then with the squarefree kernel $w$ we can write $$k= (p_1 p_2 p_3) \cdot \left(p_1^{b_1-1}p_2^{b_2-1}p_3^{b_3-1}\right) = w \cdot m$$ and then the root $x$ must have the form $$ x=w \cdot v \qquad \qquad \text{ where also } v \lt m$$
(by the last observation, a nontrivial $k$ cannot be squarefree because $m$ must be greater than $1$: $1 \le v \lt m$)
3) I hoped to combine the two empirical results somehow, such that $v \approx m/c $but don't see any further useful property except the likely irrationality of the root - so also I do not know, whether it is at all worth the effort to actually prove observation 1).
<hr>
Here is a list of the roots, scaled by the scaling factor and the integer value $3$ and the fractional value separated:
$$\small \begin{array} {r|l}
k & {\rho_k \over c} & {\rho_k \over c}=k+{1.5 \over c}+ \epsilon \\
\hline
4 & 4 + 4.16684831867 & 4 + 1.5/c + 0.344643086811 \\
5 & 5 + 3.65764105650 & 5 + 1.5/c + 0.119219557064 \\
6 & 6 + 3.48606997990 & 6 + 1.5/c + 0.0432658922944 \\
7 & 7 + 3.4241172248 & 7 + 1.5/c + 0.0158397148376 \\
8 & 8 + 3.4015117133 & 8 + 1.5/c + 0.00583236702541 \\
9 & 9 + 3.3933253612 & 9 + 1.5/c + 0.00220830950877 \\
10 & 10+3.3903889002 & 10 + 1.5/c + 0.000908352824271 \\
11 & 11+3.3893375033 & 11 + 1.5/c + 0.000442904593529 \\
12 & 12+3.3889540764 & 12 + 1.5/c + 0.000273163432055 \\
13 & 13+3.3888054311 & 13 + 1.5/c + 0.000207358901590 \\
14 & 14+3.3887395158 & 14 + 1.5/c + 0.000178178523631 \\
15 & 15+3.3887034724 & 15 + 1.5/c + 0.000162222283615 \\
\ldots & \ldots \\
497 & 497+3.388349245 & 497 + 1.5/c + 0.00000540736229828 \\
498 & 498+3.388349220 & 498 + 1.5/c + 0.00000539654486254\\
499 & 499+3.388349196 & 499 + 1.5/c + 0.00000538577062079\\
500 & 500+3.388349172 & 500 + 1.5/c + 0.00000537503931484\\
\end{array}
$$
<hr> [update]
Here is another list, where the exponent $k$ is in exponential progression and the remaining ${\epsilon \over c}$ (from the above table) is displayed by its base-$2$-logarithm:
$$\small{ \begin{array} {rr|l}
j & k=2^j & \log_2 (\epsilon /c) \\
\hline\\
2 & 4 & -0.361210135710 \\
3 & 8 & -6.24608789212 \\
4 & 16 & -11.5139506158 \\
5 & 32 & -12.4418890639 \\
6 & 64 & -13.4004277751 \\
7 & 128 & -14.3795231280 \\
8 & 256 & -15.3690269163 \\
9 & 512 & -16.3637677898 \\
10 & 1024 & -17.3611354651 \\
11 & 2048 & -18.3598186117 \\
12 & 4096 & -19.3591600122 \\
13 & 8192 & -20.3588306692 \\
14 & 16384 & -21.3586659868 \\
15& 32768 & -22.3585836430\\
\end{array} }$$
*(The range of $k$ was not continuously computed, only at the explicite values)*