$\newcommand\ka\kappa\renewcommand{\R}{\mathbb R}\newcommand{\de}{\delta}\newcommand\ep\varepsilon$Take any nonzero $a=(a_1,\dots,a_n)\in\R^n$. We have
\begin{equation*}
K=\inf_{x\in\R^n}\ka(x),\quad \ka(x):=\ka_a(x):=(\|a-x\|_2+t\|x\|_1). \tag{1}\label{1}
\end{equation*}
Since the norms $\|\cdot\|_p$ are orthant-symmetric, without loss of generality (wlog) $a_i\ge0$ for all $i\in[n]:=|[1,\dots,n\}$. Since the function $\ka$ is continuous and $\ka(x)\to\infty$ as $\|x\|_2\to\infty$, the infimum in \eqref{1} is attained at some $x=(x_1,\dots,x_n)\in\R^n$.
In what follows, (unless specified otherwise) let $x$ be such a minimizer.
If now $x_j<0$ for some $j\in[n]$ then, recalling that $a_i\ge0$ for all $i\in[n]$ and replacing the $j$th coordinate of $x$ by $-x_j$, we get another minimizer of $\ka$. So, wlog $x_j\ge0$ for all $j\in[n]$. Let
\begin{equation*}
J:=\{j\in[n]\colon x_j>0\}.
\end{equation*}
Then, differentiating the function $\ka$ at the minimizer $x$, we get
\begin{equation*}
a_j-x_j=ct\quad\text{for all}\quad j\in J,
\end{equation*}
with $x_j=0$ for $j\notin J$, where
\begin{equation*}
c:=\|a-x\|_2=\sqrt{kc^2t^2+\sum_{j\notin J}a_j^2},
\end{equation*}
whence
\begin{equation*}
c=\sqrt{\frac{\sum_{j\notin J}a_j^2}{1-kt^2}},
\end{equation*}
where
\begin{equation*}
k:=|J|,
\end{equation*}
the cardinality of $J$; here it is assumed that $k<1/t^2$.
Thus, letting
\begin{equation*}
A_1:=\sum_{j\in J}a_j,\quad A_2:=\sum_{j\notin J}a_j,\quad
B_1:=\sqrt{\sum_{j\in J}a_j^2},\quad B_2:=\sqrt{\sum_{j\notin J}a_j^2},
\end{equation*}
after some algebra we get
\begin{equation*}
K=\sqrt{1-kt^2}B_2+tA_1,\quad M=\min(\sqrt{B_1^2+B_2^2},tA_1+tA_2).
\end{equation*}
Take now any real $\ep\in(0,1)$ and consider the case when
\begin{equation*}
t\le\frac{1-\ep}{\sqrt n}; \tag{2}\label{2}
\end{equation*}
note that then $k\le n\le(1-\ep)^2/t^2$, so that the condition $k<1/t^2$ is satisfied.
Suppose now that $K<<M$; we write $A<<B$ or, equivalently, $B>>A$ if $A=o(B)$, and we write $A\ll B$ or, equivalently, $B\gg A$ if $A=O(B)$. Then
\begin{equation*}
tA_1\le K<< M\le tA_1+tA_2,
\end{equation*}
so that $tA_1<<tA_2$ and $tA_1+tA_2\ll tA_2$. Also, $A_2\le B_2\sqrt{n-k}\le B_2\sqrt n$ and hence
$tA_2\le\frac{1-\ep}{\sqrt n}B_2\sqrt n\le B_2$. So,
\begin{equation*}
M\le tA_1+tA_2\ll tA_2\ll B_2.
\end{equation*}
On the other hand, \eqref{2} implies that $kt^2\le nt^2\le(1-\ep)^2$ and hence $1-kt^2\ge1-(1-\ep)^2>0$, so that
\begin{equation*}
K\gg B_2+tA_1\ge B_2.
\end{equation*}
We conclude that, in the case \eqref{2}, the assumption $K<<M$ leads to $K\gg M$. Thus,
\begin{equation*}
t\le\frac{1-\ep}{\sqrt n}\implies K\gg M. \tag{3}\label{3}
\end{equation*}
Consider finally the case when, for some real $\ep>0$,
\begin{equation*}
1>>t\ge\frac{1+\ep}{\sqrt n}. \tag{4}\label{4}
\end{equation*}
For all $j\in[n]$, let then
\begin{equation*}
a_j:=1(j\le m)+b\,1(j>m),\quad x_j:=1(j\le m)(1-tC),
\end{equation*}
where
\begin{equation*}
m:=\Big\lceil\frac1{t^2}\Big\rceil-1,\quad C:=b\sqrt{\frac{n-m}{1-mt^2}},
\end{equation*}
and a real $b$ varies with $n$ and $t$ so that
\begin{equation*}
\frac{tm}{\sqrt n}<<b<<\frac m{\sqrt n}. \tag{5}\label{5}
\end{equation*}
Note that $m>>1$,
\begin{equation*}
n-m\gg n \tag{6}\label{6}
\end{equation*}
by \eqref{4},
\begin{equation*}
1-mt^2\le t^2, \tag{7}\label{7}
\end{equation*}
and
\begin{equation*}
b>>\frac{tm}{\sqrt n}\ge(1+\ep)\frac mn\ge\frac mn \tag{8}\label{8}
\end{equation*}
by \eqref{5} and \eqref{4}.
Next,
\begin{equation*}
K\le\ka_a(x)=K_1+K_2,\quad M=\min(M_1,M_2), \tag{9}\label{9}
\end{equation*}
where
\begin{equation*}
K_1:=\sqrt{1-mt^2}\, b\sqrt{n-m},\quad K_2:=tm,
\end{equation*}
\begin{equation*}
M_1:=\sqrt{m+(n-m)b^2},\quad M_2:=tm+t(n-m)b.
\end{equation*}
Further,
\begin{equation*}
K_1\le tb\sqrt{n-m}<<b\sqrt{n-m}\le M_1 \tag{10}\label{10}
\end{equation*}
by \eqref{7} and \eqref{4};
\begin{equation*}
K_2<< b\sqrt n\ll \sqrt{(n-m)b^2}\le M_1 \tag{11}\label{11}
\end{equation*}
by \eqref{5} and \eqref{6};
\begin{equation*}
K_1\le tb\sqrt{n-m}<<t(n-m)b\le M_2 \tag{12}\label{12}
\end{equation*}
by \eqref{7} and \eqref{6};
\begin{equation*}
K_2=tm<< tbn\ll tb(n-m)\le M_2 \tag{13}\label{13}
\end{equation*}
by \eqref{8} and \eqref{6}.
It follows from \eqref{9}--\eqref{13} that $K<<M$ in the case \eqref{4}.
Summarizing, for all $t=t_n>0$ we have
\begin{equation}
\inf_{a\in\R^n\setminus\{0\}}R_{n,t_n}(a)\to0\quad\text{if}\quad 1>>t_n\ge\frac{1+\ep}{\sqrt n}
\end{equation}
and
\begin{equation}
\inf_{a\in\R^n\setminus\{0\}}R_{n,t_n}(a)\asymp1 \quad\text{if}\quad 0<t_n\le\frac{1-\ep}{\sqrt n}
\quad\text{or}\quad t_n\gg1.
\end{equation}
$\newcommand{\Om}{\Omega}\newcommand{\Th}{\Theta}\newcommand{\B}{\mathscr B}\newcommand{\M}{\mathcal M}\newcommand\ep\varepsilon\newcommand{\de}{\delta}\newcommand{\R}{\mathbb R}$Take any $\mu\in\M(\Om)$, any open subset $\Th$ of $\Om$, and any real $\ep>0$. Let $\de:=\ep/4$.
By the Hahn decomposition theorem, there is a partition of $\Om$ into Borel sets $D^\pm$ such that $D^+$ is a positive set for $\mu$ and $D^-$ is a negative set for $\mu$.
Let
\begin{equation*}
A^\pm:=\Th\cap D^\pm. \tag{1}\label{1}
\end{equation*}
Since $|\mu|$ is inner regular, there exist compact sets
\begin{equation*}
K^\pm\subseteq A^\pm\text{ such that }|\mu|(A^\pm\setminus K^\pm)<\de. \tag{2}\label{2}
\end{equation*}
Since $\Om$ is normal, there exist open subsets $U^\pm$ of $\Th$ such that
\begin{equation*}
U^\pm\supseteq K^\pm\text{ and }U^+\cap U^-=\emptyset. \tag{3}\label{3}
\end{equation*}
Since the sets $K^\pm$ are compact and $\Om$ is locally compact, without loss of generality the closures of the sets $U^\pm$ are compact.
By Urysohn'slemma, there exist continuous functions $f^\pm\colon\Om\to\R$ such that
\begin{equation*}
0\le f^\pm\le1,\quad f^\pm=1\text{ on }K^\pm,\quad f^\pm=0\text{ on }\Om\setminus U^\pm. \tag{4}\label{4}
\end{equation*}
Let
\begin{equation*}
f:=f^+-f^-.
\end{equation*}
Then $f^+f^-=0$, whence $|f|\le1$. Also, $f=0$ on $\Om\setminus(U^+\cup U^-)$. So, recalling that the closures of the sets $U^\pm$ are compact, we see that $f\in C_c(\Om)$. Also, since $U^\pm$ are subsets of $\Th$, we have $|f|\le1_\Th$.
It remains to show that
\begin{equation*}
\int_\Om f\,d\mu\ge|\mu|(\Th)-\ep. \tag{$*$}\label{*}
\end{equation*}
To do this, note that, by \eqref{3}, \eqref{2}, and \eqref{1},
\begin{equation}
\begin{aligned}
|\mu|(U^-\setminus K^-)&\le|\mu|(\Th\setminus U^+\setminus K^-) \\
&=|\mu|(\Th)-|\mu|(U^+)-|\mu|(K^-) \\
&\le|\mu|(\Th)-|\mu|(K^+)-|\mu|(K^-) \\
&<|\mu|(\Th)-|\mu|(A^+)-|\mu|(A^-)+2\de=2\de.
\end{aligned}
\tag{5}\label{5}
\end{equation}
So, by \eqref{4}, \eqref{3}, \eqref{2}, \eqref{5}, and \eqref{1},
\begin{equation*}
\begin{aligned}
\int_\Om f\,d\mu&=\int_{U^+} f^+\,d\mu-\int_{U^-} f^-\,d\mu \\
&\ge\int_{K^+} f^+\,d\mu-\int_{K^-} f^-\,d\mu -\int_{U^-\setminus K^-} f^-\,d\mu \\
&=\mu(K^+)-\mu(K^-) -\int_{U^-\setminus K^-} f^-\,d\mu \\
&\ge\mu(K^+)-\mu(K^-) -|\mu|(U^-\setminus K^-) \\
&>\mu(A^+)-\de-\mu(A^-)-\de -2\de \\
&=|\mu|(\Th)-4\de=|\mu|(\Th)-\ep,
\end{aligned}
\end{equation*}
so that \eqref{*} is proved. $\quad\Box$
Best Answer
This is just a standard application of the Baire category theorem.
Proposition: Suppose that $X$ is a topological space where the Baire category theorem holds and $g_{n}:X\rightarrow[0,\infty]$ for each natural number $n$. Suppose that $\overline{\lim}_{n}\sup_{I}g_{n}\geq 1$ for each non-empty open set $I$. Then there is a point $x\in X$ such that $\overline{\lim}_{n}g_{n}(x)\geq 1$.
Proof: If $0<\delta<1$ and $n$ is a natural number, then let $U_{n,\delta}=g_{n}^{-1}(\delta,\infty)$. Then each $U_{n,\delta}$ is open. Let $O_{n,\delta}=\bigcup_{k=n}^{\infty}U_{k,\delta}$. Then $O_{n,\delta}$ is open and dense. Therefore, $\bigcap_{m,n}O_{n,1-1/m}$ is dense by the Baire category theorem. If $x\in\bigcap_{m,n}O_{n,1-1/m}$, then for each $m>0$, we have $x\in O_{n,1-1/m}$ for all $n$. Therefore, since $x\in O_{n,1-1/m}$, we know that $x\in U_{k,1-1/m}$ for infinitely many $k$. Thus, $g_{k}(x)>1-1/m$ for infinitely many $k$. Therefore, $\overline{\lim}_{n\rightarrow\infty}g_{n}(x)\geq 1-1/m$, so since $m$ is arbitrary, we conclude that $\overline{\lim}_{n\rightarrow\infty}g_{n}(x)\geq 1$ for $x\in\bigcap_{m,n}O_{n,1-1/m}$. Q.E.D.