We have $S = \lbrace 2, \ell \rbrace$ and a residual representation $\bar{r} \colon G_{\mathbb{Q},S} \to \mathrm{GL}_ 2(\mathbb{F})$ with a universal deformation over $R^\mathrm{univ}$ under the conditions (A) $\det r = \epsilon_\ell^{-1}$, (B) some condition on $r \vert_{G_{\mathbb{Q}_ 2}}$, (C) some condition on $r \vert_{G_{\mathbb{Q}_ \ell}}$.
We looked at the minimum number of generators $d = \dim H_\mathscr{L}^1(G_{\mathbb{Q},S}, \mathrm{ad}^0 \bar{r})$ and $$ 0 \to I \to \mathcal{O}[[x_1, \dotsc, x_d]] \twoheadrightarrow R^\mathrm{univ} $$ and tried to understand $\Hom_\mathbb{F}(I/\mathfrak{m} I, \mathbb{F})$. Given any nonzero element $f$, we could look at $\ker f = J_f / \mathfrak{m} I$ where $f \colon I/J_f \cong \mathbb{F}$. Because $d$ is minimal, there does not exist a lift of $r^\mathrm{univ}$ to $\mathcal{O}[[x]] / J_f$. We were looking at what constraint it puts.
There exists a set-theoretic lift $$ \tilde{r} \colon G_{\mathbb{Q},S} \to \mathrm{GL}_ 2(\mathcal{O}[[x_1, \dotsc, x_d]] / J_f) $$ satisfying (A), and from this we extracted a cocycle $$ \phi(\sigma, \tau) = f(\tilde{r}(\sigma) \tilde{r}(\tau) \tilde{r}(\sigma \tau)^{-1} - 1) \in \mathrm{ad}^0 \bar{r}. $$ Changing the lifting changes $\phi$ by a coboundary, so we got a canonical class $$ [\phi] \in H^2(G_{\mathbb{Q},S}, \mathrm{ad}^0 \bar{r}). $$ We then see that $\bar{r}$ lifts to $\tilde{r}$ if and only if $[\phi] = 0$.
Local obstructions
Claim 1. For $v = 2$ or $v = \ell$, there exists a lift $\hat{r}_ v \colon G_{\mathbb{Q}_ v} \to \mathrm{GL}_ 2(\mathcal{O}[[x_1, \dotsc, x_d]] / J_f)$ lifting $r^\mathrm{univ} \vert_{G_{\mathbb{Q}_ v}}$ satisfying (B) and (C) respectively.
This is a special feature of the local conditions. For $v = 2$, we are looking at the condition that $$ r \vert_{G_{\mathbb{Q}_ 2}} \sim \begin{pmatrix} 1 & \psi \epsilon_\ell^{-1} \br 0 & \epsilon_\ell^{-1} \end{pmatrix} \otimes \delta, \quad \delta^2 = 1 $$ where we have $\psi \in Z_\mathrm{cts}^1(G_{\mathbb{Q}_ 2}, R(\epsilon_ \ell))$. So it really suffice to show that $$ Z_\mathrm{cts}^1(G_{\mathbb{Q}_ 2}, \mathcal{O}[[x_1, \dotsc, x_d]]/J_f(\epsilon_ \ell)) \twoheadrightarrow Z_\mathrm{cts}^1(G_{\mathbb{Q}_ 2}, R^\mathrm{univ}(\epsilon_\ell)) $$ is surjective. It is sufficient to prove that $$ H_\mathrm{cts}^1(G_{\mathbb{Q}_ 2}, \mathcal{O}[[x_1, \dotsc, x_d]] / J_f(\epsilon_\ell)) \twoheadrightarrow H_\mathrm{cts}^1(G_{\mathbb{Q}_ 2}, R^\mathrm{univ}(\epsilon)) $$ is surjective, because we can always lift coboundaries due to the fact that $\mathcal{O}[[x_1, \dotsc, x_d]] / J_f \to R^\mathrm{univ}$ is surjective.
Now using Kummer theory, we can identify this with $$ \widehat{\mathbb{Q}_ 2^\times} \otimes_{\hat{\mathbb{Z}}} \mathcal{O}[[x_1, \dotsc, x_d]]/J_f \to \widehat{\mathbb{Q}_ 2^\times} \otimes_{\hat{\mathbb{Z}}} R^\mathrm{univ}. $$ Because $\mathbb{Z}_2^\times$ is a pro-$2$-group $2$ is invertible in the other rings, this can be identified under the valuation map with $$ \mathcal{O}[[x_1, \dotsc, x_d]] / J_f \twoheadrightarrow R^\mathrm{univ}, $$ which is indeed surjective.
Let us now consider the case of $v = \ell$. What we want is $$ \hat{r}_ \ell \colon G_{\mathbb{Q}_ \ell} \to \mathrm{GL}_ 2(\mathcal{O}[[x_1, \dotsc, x_d]] / J_f) $$ satisfying $\det \hat{r}_ \ell = \epsilon_\ell^{-1}$ and $\hat{r}_ \ell$ lifts to $r^\mathrm{univ} \vert_{G_{\mathbb{Q}_ \ell}}$ and for all open $K \subseteq \mathcal{O}[[x_1, \dotsc, x_d]]$ we have that $\hat{r}_ \ell \bmod{K}$ has image in $\mathbb{G}$. We can reduce to the Artinian setting and show the following general fact.
Claim 2. Let $R \in \mathcal{C}_ \mathcal{O}$ be Artinian and let $K \subseteq R$ be an ideal. Given $r \colon G_{\mathbb{Q}_ \ell} \to \mathrm{GL}_ 2(R/K)$ satisfying conditions (A) and (C), there exists a lift $\tilde{r} \colon G_{\mathbb{Q}_ \ell} \to \mathrm{GL}_ 2(R)$ satisfying conditions (A) and (C).
Proof.
We find $M$ such that $r = \mathbb{G}(M)$, where $\mathrm{Fil}^i M$ has rank $2-i$ for $0 \le i \le 2$. We choose a basis $e_1, e_2$, so that $$ \Phi^1\begin{pmatrix} 1 \br 0 \end{pmatrix} = \begin{pmatrix} a \br b \end{pmatrix}, \quad \Phi^0\begin{pmatrix} 1 \br 0 \end{pmatrix} = \begin{pmatrix} \ell a \br \ell b \end{pmatrix}, \quad \Phi^0\begin{pmatrix} 0 \br 1 \end{pmatrix} = \begin{pmatrix} c \br d \end{pmatrix}, \quad \begin{pmatrix} a & c \br b & d \end{pmatrix} \in \mathrm{GL}_ 2(R/K). $$ In fact, if we look at the determinant condition, we see that $\det r$ is the character that is $\epsilon_\ell^{-1}$ times the unramified character $\mathrm{Frob} \mapsto \ell (ad-bc)$. This shows that $$ \begin{pmatrix} a & c \br b & d \end{pmatrix} \in \mathrm{SL}_ 2(R/K). $$ Now we use the fact that $\mathrm{SL}_ 2(R) \twoheadrightarrow \mathrm{SL}_ 2(R/K)$ is surjective.
The conditions (B) and (C)
Let us now go back to the global situation and the set-theoretic lift $$ \tilde{r} \colon G_{\mathbb{Q},S} \to \mathrm{GL}_ 2(\mathcal{O}[[x_1, \dotsc, x_d]] / J_f) $$ satisfying (A). We were looking at the cocycle $$ \phi(\sigma, \tau) = f(\tilde{r}(\sigma) \tilde{r}(\tau) \tilde{r}(\sigma \tau)^{-1} - 1). $$ But we have our local lifts $\hat{r}_ 2$ and $\hat{r}_ \ell$, so we can compare $$ \psi_2(\sigma) = f(\tilde{r}(\sigma) \hat{r}_ 2(\sigma)^{-1} - 1), \quad \psi_\ell(\sigma) = f(\tilde{r}(\sigma) \hat{r}_ \ell(\sigma)^{-1} - 1), $$ for $\sigma \in G_{\mathbb{Q}_ 2}$ and $\sigma \in G_{\mathbb{Q}_ \ell}$ respectively. We checked that $\partial \phi = 0$, and we can now check that $$ \phi \vert_{G_{\mathbb{Q}_ v}} = \partial \psi_v. $$
There was also some ambiguity in choosing $\tilde{r}$. For $\beta \in C^1(G_{\mathbb{Q},S}, \mathrm{ad}^0 \bar{r})$ we could change $$ \tilde{R} \mapsto (1 - f^{-1} \beta(\sigma)) \tilde{r}(\sigma), \quad \phi \mapsto \phi + \partial \beta, \quad \psi_v \mapsto \psi_v + \beta \vert_{G_{\mathbb{Q}_ v}}. $$
Conclusion 3. There exists a lift of $r^\mathrm{univ}$ to $\mathcal{O}[[x_1, \dotsc, x_d]] / J_f$ satisfying (A), (B), (C) precisely if we can choose $\beta$ such that
- $\phi + \partial \beta = 0$,
- $\psi_v + \beta \vert_{G_{\mathbb{Q}_ v}} \in \tilde{L}_ v \in Z^1(G_{\mathbb{Q}_ v}, \mathrm{ad}^0 \bar{r})$.
So what we are looking at is the complex $$ \begin{align} C^1(G_{\mathbb{Q},S}, \mathrm{ad}^0 \bar{r}) &\to C^2(G_{\mathbb{Q},S}, \mathrm{ad}^0 \bar{r}) \oplus \bigoplus_{v \in S} C^1(G_{\mathbb{Q}_ v}, \mathrm{ad}^0 \bar{r}) / \tilde{L}_ v \br &\to C^3(G_{\mathbb{Q},S}, \mathrm{ad}^0 \bar{r}) \oplus \bigoplus_{v \in S} C^2(G_{\mathbb{Q}_ v}, \mathrm{ad}^0 \bar{r}), \end{align} $$ where the maps are $$ \beta \mapsto (\partial \beta, (\beta \vert_{G_{\mathbb{Q}_ v}})), \quad (\phi, (\psi_v)) \mapsto (\partial \phi, (\phi \vert_{G_{\mathbb{Q}_ v}} - \partial \psi)). $$ We define $H_\mathscr{L}^2(G_{\mathbb{Q},S}, \mathrm{ad}^0 \bar{r})$ to be the middle cohomology of this complex.
Proposition 4. We have an injection $$ \Hom_\mathbb{F}(I/\mathfrak{m}I, \mathbb{F}) \hookrightarrow H_\mathscr{L}^2(G_{\mathbb{Q},S}, \mathrm{ad}^0 \bar{r}). $$
So from this we will get $$ \dim R^\mathrm{univ} \le 1 + \dim H_\mathscr{L}^1(G_{\mathbb{Q},S}, \mathrm{ad}^0 \bar{r}) - \dim H_\mathscr{L}^2(G_{\mathbb{Q},S}, \mathrm{ad}^0 \bar{r}). $$