Ok, if you scroll back to post #2 I sent Jason a pic of a control circuit I designed for a job I did on the same exact West ATS that you show in you links.
The circuit for the panel is shown in the drawing you sent. The PSDR (Pre Signal Delay Relay) is the Idec cube relay #RH2B-UL, 120VAC. I labeled it PSR. The PSDT (Pre Signal Delay Timer) is the Omron Timer#H3CR, 120VAC, set for 20 sec. I labeled it PST. On the terminal strip you connect 21, 24 and 18 as shown in your dwg. Terminals A, B & C are the relay contacts 4, 12 & 8 that provide the pre signal to the elevator controls. The way it works is when the signal is sent to transfer the ATS, both the relay and timer energize (pre signal sent to elev.) Then after the timer times out, the PSDT contact closes and the motor TM energizes, transferring the ATS.
If you feel more comfortable, your other option is to buy an off the shelf module made by Kohler for their ATS's but can be used on any brand (see attached files.) I think they are still available. Google Kohler part# 297124.
What sounds simple to me (51 yrs doing controls) may be confusing to others. You may have to engage an experienced electrician or controls guy to help you. Hope this helps.
View attachment 2555638View attachment 2555639 19014 Kohler elev presig dwg_294147.pdf