Don
I suggest that you build a sequencer box. There are several ways to make a sequencer, and it is an inexpensive and very worthwhile addition to your shack.
When you want to transmit, you close your push-to-talk switch... to activate the sequencer (not the Desk KW directly). One output (call it the 1st output) of the sequencer will provide a path the ground, almost immediately after you close the push-to-talk switch. You can use the 1st output to activate the antenna relay. Another output (call it the 2nd output) of the sequencer will provide a path to ground about 0.5 seconds after the first output provides a path to ground. You can use the 2nd output to turn on the Desk KW.
When you want to stop transmitting, you open the push-to-talk switch. The 2nd output's path to ground opens up almost immediately... turning off the Desk KW. The 1st output's path to ground open's up about 0.5 seconds later, deactivating the antenna relay.
I made the sequencer in my shack using old fashioned relay technology. I needed two 12V DPDT relays, and a capacitor with (roughly) 3300uF of capacitance to provide the needed delay. My sequencer provides 12V to the antenna relay from output 1; but it is easy to modify this design, to provide a path to ground at output 1.
The wiring diagrams are attached. The first attachment is the complete wiring diagram. The 2nd and 3rd attachments are partial diagrams to show (separately) the circuitry for switching the antenna relay (the 1st output) and the circuitry for switching the transmitter (the 2nd output).
In this design, you can use relays that have coils with 360 ohms of series resistance, and requiring 23.3mA of coil current to operate.
http://www.mouser.com/ds/2/418/RT2pb0411-205066.pdfWith 100 ohms in series ... for the delay ... 12 volts across 460 ohms produces 26mA of current. If 26mA of current is not quite enough... to activate the delayed-turn-on relay... then use less resistance (e.g. 47 ohms) and more capacitance (e.g. 2 x 3300uF)
The delay time will be roughly two RC time constants = 2 x 78 ohms x 3300uF = 0.5 seconds
Separately:
I don't know whether or not Dow-Key relays are reliable enough. I use vintage Dow-Key relays; and also new Tohsu relays, which are much less expensive than new Dow-Key relays. The Dow-Key relays do require occasional contact cleaning... and sometimes the SO-239 sockets on my Dow-Key relays don't make good contact with the mating PL-259 connectors. To fix that, I usually roll up a 3/8 inch wide strip of aluminum foil into a 1/8 inch diameter cylinder... and insert it into the Dow-Key SO-239 center sleeve... before attaching the PL-259 connector.
Stu