I don't think what you say is true. Let $k$ be a field, $V$ a $k$-vector space of uncountable dimension and $R=k\oplus V$ the $k$-algebra where $V$ is a square-zero ideal. Consider $\mathfrak T=D(R)$ the derived category of $R$, and $A=R$. Take a non-trivial vector $0\neq v\in V$. The complex $$B=\cdots\rightarrow 0\rightarrow R\stackrel{v}\longrightarrow R\rightarrow 0\rightarrow \cdots$$ is in $\langle A\rangle_{\aleph_1}$, in fact it is in $\langle A\rangle_{\aleph_0}$. In this case $\mathfrak T_*(A,A)=R$ concentrated in degree $0$ and $\mathfrak T_*(A,B)=R/(v)\oplus V[1]$, which is not countably generated since $\dim_kV$ is uncountable.

What you claim is true under some transfinite noetherianity hypothesis, e.g. it is proven in the literature under the hypothesis that $\mathfrak T_*(A,A)$ is countable.

I don't think what you say is true. Let $k$ be a field, $V$ a $k$-vector space of uncountable dimension and $R=k\oplus V$ the $k$-algebra where $V$ is a square-zero ideal. Consider $\mathfrak T=D(R)$ the derived category of $R$, and $A=R$. Take a non-trivial vector $0\neq v\in V$. The complex $$B=\cdots\rightarrow 0\rightarrow R\stackrel{v}\longrightarrow R\rightarrow 0\rightarrow \cdots$$ is in $\langle A\rangle_{\aleph_1}$, in fact it is in $\langle A\rangle_{\aleph_0}$. In this case $\mathfrak T_*(A,A)=R$ concentrated in degree $0$ and $\mathfrak T_*(A,B)=R/(v)\oplus V[1]$, which is not countably generated since $\dim_kV$ is uncountable.

What you claim is true under some transfinite coherence hypothesis, e.g. it is proven in the literature under the hypothesis that $\mathfrak T_*(A,A)$ is countable.

I don't think what you say is true. Let $k$ be a field, $V$ a $k$-vector space of uncountable dimension and $R=k\oplus V$ the $k$-algebra where $V$ is a square-zero ideal. Consider $\mathfrak T=D(R)$ the derived category of $R$, and $A=R$. Take a non-trivial vector $0\neq v\in V$. The complex $$B=\cdots\rightarrow 0\rightarrow R\stackrel{v}\longrightarrow R\rightarrow 0\rightarrow \cdots$$ is in $\langle A\rangle_{\aleph_1}$, in fact it is in $\langle A\rangle_{\aleph_0}$. In this case $\mathfrak T_*(A,A)=R$ concentrated in degree $0$ and $\mathfrak T_*(A,A)=R/(v)\oplus V^\perp[1]$, where $V^\perp$ is a direct sum complement of the vector subspace generated by $v\in V$. This $R$-module is not countably generated since $\dim_kV^\perp$ is uncountable.

What you claim may be true under some transfinite noetherianity hypothesis, e.g. in the literature, it is shown that your claim is true if $\mathfrak T_*(A,A)$ is countable.

I don't think what you say is true. Let $k$ be a field, $V$ a $k$-vector space of uncountable dimension and $R=k\oplus V$ the $k$-algebra where $V$ is a square-zero ideal. Consider $\mathfrak T=D(R)$ the derived category of $R$, and $A=R$. Take a non-trivial vector $0\neq v\in V$. The complex $$B=\cdots\rightarrow 0\rightarrow R\stackrel{v}\longrightarrow R\rightarrow 0\rightarrow \cdots$$ is in $\langle A\rangle_{\aleph_1}$, in fact it is in $\langle A\rangle_{\aleph_0}$. In this case $\mathfrak T_*(A,A)=R$ concentrated in degree $0$ and $\mathfrak T_*(A,B)=R/(v)\oplus V[1]$, which is not countably generated since $\dim_kV$ is uncountable.

What you claim is true under some transfinite noetherianity hypothesis, e.g. it is proven in the literature under the hypothesis that $\mathfrak T_*(A,A)$ is countable.