  We consider the Forrester-Baxter RSOS lattice models with crossing parameter
$\lambda=(m'\!-\!m)\pi/m'$ in Regime~III. In the continuum scaling limit, these
models are described by the minimal models ${\cal M}(m,m')$. We conjecture
that, for $\lambda<\pi/n$, the $n\times n$ fused RSOS models with $n\ge 2$ are
described by the higher-level coset $(A^{(1)}_1)_k\otimes
(A^{(1)}_1)_n/(A^{(1)}_1)_{k+n}$ at fractional level $k=nM/(M'\!-\!M)-2$ with
$(M,M')=\big(nm-(n\!-\!1)m',m'\big)$. To support this conjecture, we
investigate the one-dimensional sums arising from Baxter's off-critical corner
transfer matrices. In unitary cases ($m=m'\!-\!1$) it is known that, up to
leading powers of $q$, these coincide with the branching functions
$b_{r,s,\ell}^{m'\!-n,m'\!,n}(q)$. For general nonunitary cases ($m<m'\!-\!1$),
we identify the ground state one-dimensional RSOS paths and relate them to the
quantum numbers $(r,s,\ell)$ in the various sectors. For $n=1,2,3$, we obtain
the local energy functions $H(a,b,c)$ in a suitable gauge and verify that the
associated one-dimensional sums produce finitized forms that converge, as $N$
becomes large, to the fractional level branching functions
$b_{r,s,\ell}^{M,M'\!,n}(q)$. Extending the work of Schilling, we also
conjecture finitized bosonic branching functions
$b_{r,s,\ell}^{M,M'\!,n;(N)}(q)$ for general $n$ and check that these agree
with the one-dimensional sums for $n=1,2,3$ out to system sizes $N=14$. Lastly,
the finitized Kac characters $\chi_{r,s,\ell}^{P,P'\!,n;(N)}(q)$ of the
$n\times n$ fused logarithmic minimal models ${\cal LM}(p,p')$ are obtained by
taking the {\em logarithmic limit\/} $m,m'\to\infty$ with $m/m'\to p/p'+$.
