Transversal temperature pattern formation has been observed in laboratory and industrial catalytic packed-bed reactors (PBRs) used for conducting exothermic reactions. These patterns or nonuniform states can strongly affect reactor performance and pose severe safety issues. Recent studies show that symmetry-breaking bifurcations may cause transversal pattern formation in a reactor operated under nonadiabatic conditions. In this study, we show that wall temperature, which dictates the instantaneous and overall heat exchange rate, strongly influences the selection and dynamics of various target and rotating patterns exhibited in a shallow nonadiabatic PBR. We demonstrate this by linear stability analysis guided extensive numerical simulations of a shallow reactor model assuming periodic blocking-reactivation kinetics for the catalytic reaction. Transversal spatiotemporal patterns predicted in lab-scale (similar to 6 cm diameter) and/or bench-scale (similar to 60 cm) reactors, include rotating patterns, inward/outward/multi-ring targets, quasi-stationary moving patterns, and symmetric and asymmetric spirals. We show that wall temperature modulates the transition between these targets and rotating transversal nonuniform states at both scales. We argue that rich and intricate patterns observed much more in bench-scale reactors than those in lab scale reactors are possibly due to reduction in the heat removal time upon increase in diameter by 10-fold. We further classify the simulated transversal patterns into three regimes, viz., (i) heating, (ii) heating and cooling, and (iii) cooling, based on the nature of wall heat exchange rate dynamics dictated by the (instantaneous) local temperature near the reactor wall. Wall heat exchange rate dynamics being an experimental observable makes it a signature of a nonuniform state present inside the reactor.