Are the claims/proof sketch made in the next section valid?
My work
I've become interested in other ways of (ultimately) constructing the real numbers.
Consider the ring $\mathcal R$ of Laurent polynomials over the integers,
$\tag 1 \mathcal R = \{ \displaystyle{\sum_{k={-n}}^n a_k x^k} \mid a_k \in \Bbb Z \}$
Define a set $\mathcal I$ given by
$\tag 2 \mathcal I = \{ (x-2) \displaystyle{\sum_{k={-n}}^n a_k x^k} \mid \text{ where } a_k \in \Bbb Z \}$
The set $\mathcal I$ forms an ideal. the quotient
$\tag 3 \mathcal B = \frac{\mathcal R}{\mathcal I}$
contains the integers. Moreover, $2^{-1} \in \mathcal B$.
An immediate problem now is that a number in $x \in \mathcal B$ can have different representations of the form
$\tag 4 x = \displaystyle{\sum_{k={-n}}^n a_k 2^k}\quad \text{where } a_k \in \Bbb Z$
Lemma 1: Let $x \in \mathcal B$ satisfy $\text{(4)}$. Then an algorithm can be applied transforming $x$ into the form
$\tag 5 x = \displaystyle{\sum_{k={-m}}^m b_k 2^k}\quad \text{where } |b_k| \lt 2$
Lemma 2: Let $x \in \mathcal B$ satisfy $\text{(5)}$ with $b_m \gt 0$. Then an algorithm can be applied transforming $x$ into the form
$\tag 6 x = \displaystyle{\sum_{k={p}}^q c_k 2^k}\quad \text{where } p \le q \land c_k \in \{0,1\}$
Lemma 3: Let $x \in \mathcal B$ satisfy $\text{(5)}$ with $c_p \ne 0$ and $c_q \ne 0$.
Then all the coefficients $c_p, \dots, c_q$ are uniquely determined.
Proof
If we have two representations then using an argument similar to the one found here can be used to show that $x-2$ can't divide the difference so by ideal theory we have uniqueness. $\quad \blacksquare$
Theorem 4: Every number nonzero number $x \in \mathcal B$ has one and only representation, either in the form
$\tag 7 x = \displaystyle{\sum_{k={p}}^q c_k 2^k}\quad \text{where } p \le q \land c_p = +1 \land c_q = +1 \land c_k \in \{0,+1\}$ xor $\tag 8 x = \displaystyle{\sum_{k={p}}^q c_k 2^k}\quad \text{where } p \le q \land c_p = -1 \land c_q = -1 \land c_k \in \{0,-1\}$
Moreover, $\mathcal B$ is naturally endowed with a dense total ordering.
Note: The proofs are similar no matter what 'base' is chosen. So you can also construct, say, decimal fractions in this manner.
The Wikipedia article Construction of the real numbers gives several constructions. The specific one you suggest seems to be somewhat related to an approach referred to in reference 8 "The Real Numbers as a Wreath Product" by F. Faltin, N. Metropolis, B. Ross and G.-C. Rota. This is discussed in the MathOverflow question 339348 "The real numbers as a wreath product?" with an answer giving some insight into the construction. The main problem there seems to be how to handle required "carries" to ensure uniqueness.
You only deal with a finite number of digits, but this is somewhat related to MSE question 1437992 "Multiplication of doubly-infinite decimal numbers" whose answer gives a link to the original "Wreath product" article.