I totally feel you - the one reservation is that I’m using (gasp) not the original Minivac Relay sound. I need to go to a Computer Museum that will let me record one to offer a more authentic simulated experience.
So tldr - let me get a clean real sound and then we can come back to this question :)
You may not need to get a museum's cooperation. After a quick search it seems these come up for sale from time to time. I saw one on the Vintage Computer Federation forums offered in 2021 for $500. You may be able to find a collector online who has one and will record the relays for you. A lot of folks have pretty high-quality USB mics for conferencing/podcasting these days.
An alternative might be going to a regional VCF show where one will be (possibly by pre-arrangement with a collector). There are five or six shows held around the country every year. Unlike a museum almost all the devices there are available for hands-on interaction.
The relay click sound you're hearing should only happen when a relay turns on, and when it turns off. There is no sound when a button is pressed - you can check this by loading an empty simulator and pressing any of the buttons at the bottom (the simulator won't make any sound):
Feature request: I was wanting for a bit more sound… I mean, when I see "relay computer", I'm heading there for lots of soft clicks.