Is this derivation of Black Hole entropy viable?

This question is motivated by this one.

Suppose $l$ is the minimum measurable unit of length. What is entropy of a spinless particle contained in this interval?

We know that entropy of a two-level system depends on the probabilities of the respective levels, if the probability of the state 0 is $p_0$, then the entropy (in natural units) is:

$$S= -\sum_{i=0}^1 p_i \ln p_i = -p_0 \ln p_0 - (1 - p_0) \ln (1 - p_0)$$

So if $p_0=1/2$ then $S=\ln 2\: \mathrm{nat}$, equal to $1\: \mathrm{bit}$. A particle which has the maximum in the middle has entropy of $1\: \mathrm{bit}$ (it is equally likely to be measured to the right and to the left of the middle).

Since we cannot measure intervals smaller than $l$, we cannot make guesses about where the maximum of the probability for the particle is located. As such, if we assume that the particle is equally likely have the maximum of the probability in any point on the interval $x\in[0,l]$, the total entropy becomes $$S=\int_0^l \frac{-(1-\frac xl) \ln (1- \frac xl)-\frac xl \ln (\frac xl)}{l} \, dx=\int_0^1 -(1-x) \ln (1-x)-x \ln (x) \, dx=\frac12$$

An entropy of a similar particle contained in a square area with side $l$ will be twice more, that is $1\: \mathrm{nat}$.

Now if we assume that $l=2l_p$ where $l_p$ is the Planck length, we arrive that such spinless particle has entropy of $1\: \mathrm{nat}$ per 4 square Planck length or $1/4\: \mathrm{nat}$ for one square Planck length.

Thus from the only assumption that double Planck length is the minimum measurable interval, and double Planck length squared is expected to contain 1 particle on average we arrive at the standard value of the Black Hole entropy in nats:

$$S=\frac{A}{4l_p^2}=\frac14 A_p$$

Where $A_p$ is the area in Planck units.

Sometimes I encountered a claim that the fundamental unit of information is 1 bit. From the above considerations it follows that possibly the fundamental unit is 1/2 (or 1 or 1/4) nat.

UPDATE

Note that the distance of $2l_p$ between two particles is natural if we assume that the particles are planckons, whose radius is Planck length $l_p$. As such, the Black Hole can be viewed as a spherical shell consisting of one layer of planckons.