Found the cause of the clicking sound (at least, a first step to confirming it). Using ukrkoz’s frive shaft test I found the driver’s side from drive shaft has play at the transmission end: I can also hear it knocking when I move it.

The dealer’s diagnosis of bad motor mounts was either wrong or he wanted to rip me off. The motor mounts look good, including the rear torque absorber. The driver side from motor mount seems to torque a little more than others but I can’t say that it’s out of spec. no clunking nor knocking of the engine over pot holes, manhole covers, etc. And they seem pretty solid when I tried applying torque to them with a pry bar. The dealer’s diagnosis of bad motor mounts was either wrong or he wanted to rip me off.

I’ll take it one atep at a time and replace a known bad part: the driver side drive shaft as I might very well have been the cause of the defect last year when replacing from wheel bearings.

It’ll take me a couple of weeks or so to get to the job but I’ll report back if replacing that one drive shaft resolved the clicking sound while braking.

Thanks a bunch for leading me in the right direction.

Update on the question of bad engine mounts on my 2007 RX400H. Well, ukrkoz was right: steering wheel vibration when accelerating to merge into highway traffic was in fact NOT due to bad engine mounts but due to a bad CV drive axle.

Yesterday, I replaced the driver side front drive axle that I had separated by accident last year: clicking noise when braking and steering wheel vibration when accelerating are both gone.

Regarding the engine mounts, I raised the car on a lift at a DIY place yesterday (where I replaced the drive axle) so had lots of room and maneuverability to test the mounts. They’re in fact worn but nothing that will make the engine fall off anytime soon. And certainly not enough wear to make the steering wheel vibrate or the clicking sound when braking

So, ukrkoz, thanks a bunch for asking me what I the problem was and why I concluded that the engine mounts were bad: You helped me rethink my diagnosis. I certainly will replace the engine mounts in the spring, after the winter, as it can wait. You helped me pinpoint the problem and prevented me from just throwing parts at a misdiagnosed issue. Well done!!