If you have a Pocket Wizard (or any other type of wireless trigger) attached to the 430EX III-RT you need to take it out of wireless slave mode. In wireless slave mode the flash is ignoring the hot foot and "listening" to its internal 'RT' radio receiver for instructions. But your Pocket Wizard radio transmitter does not speak 'RT', it speaks 'PW'. The two different protocols are not compatible. To listen to the receiver attached to the hot foot of the flash (or the PC Terminal connection) the flash needs to be set the same way as if it were attached directly to your camera's hot shoe in manual flash mode.

If you don't have a receiver attached to the foot of the 430EX III-RT then your transmitter needs to be compatible with the 'RT' radio receiver in the flash. Canon's RT flash units (430EX III-RT and 600EX-RT) and transmitters (ST-E3-RT) or third party 'RT' units such as Yongnuo's 'RT' flashes (YN600EX-RT, YN600EX-RT II) or transmitters (YN-E3-RT) are 'RT' radios. The radios in your Pocket Wizards are not 'RT' radios, they use Pocket Wizard's own proprietary radio protocol.