  We report observations of CO($J$=9$\to$8) and OH$^+$($N$=1$\to$0) toward the
four millimeter-selected lensed starburst galaxies SPT 2354-58, 0150-69,
0314-44, and 0452-50, using the Atacama Large Millimeter/submillimeter Array
Compact Array (ALMA/ACA), as part of a larger study of OH$^+$ in the early
universe. In this work, we use these observations for the main purpose of
spectroscopic redshift measurements. For all sources except 0452-50, we confirm
the previously reported most likely redshifts, and we find typical CO and
OH$^+$ properties for massive starbursts. For 0452-50, we rule out the
previously reported value of $z$=2.0105, measuring a firm redshift of
$z$=5.0160 based on [OI], [CII], H$_2$O, and CO emission instead when adding in
ancillary ALMA data. Previously, 0452-50 was considered an outlier in relations
between dust temperature, far-infrared luminosity and redshift, which may have
hinted at an unusually cold starburst with a dust temperature of only $T_{\rm
dust}$=(21$\pm$2) K. Instead, our new measurements suggest it to be among
highly luminous massive dusty starbursts at $z$$>$5, with rather typical
properties within that population. We find a revised dust temperature of
$T_{\rm dust}$=(76.2$\pm$2.5) K, and updated lensing-corrected far-infrared and
infrared luminosities of (2.35$^{+0.09}_{-0.08}$) and
(4.07$^{+0.25}_{-0.27}$)$\times$10$^{13}$ $L_\odot$, respectively - i.e., about
an order of magnitude higher than previously reported. We thus do not find
evidence for the existence of unusually cold starburst galaxies in the early
universe that were missed by previous selection techniques.
