Good question!
I’m often taking the bike for birding. I’m also curious about tips and experiences from others.
I keep my binoculars most often around my neck when cycling, with not only rainguards, but also objective covers.
Biking is the main (only real) reason I‘v got objective covers: against splashes, dust, etc., but also just in case of some other potential impact (fall, or e.g. some protruding element of the bike like the bell hitting it when suddenly pulling the breaks due to the binoculars just being hanging around my neck etc.). I have never had any such an impact on my binoculars, but nearly. You loose no more than 1 or 2 seconds opening the objective covers when used to.
If just cycling to/from some place crossing e.g.the city or too much in a hurry to get back (havingalready birded « too long »

) I sometimes put my binoculars in the rear pannier, but first in their own bag/case and with some soft thing at the bottom or so of the cycling bag (e.g.sweater or whatever at hands).
My scope is on my back, on its tripod using a Scopac. I absorb any potential road bump

. It is very quick to setup this way.
I‘m wondering on how to put the scope and tripod on the bike without fearing the bumps/shocks, because I would take it along more often I thing.
Do you have some padding protecting your scope in the carrier, or are you only riding on very smooth roads?
I’m not doing any cyclocross, MTB or so at all, but there are road sections with cobblestones or dips in the cycling roads you might not always be able to avoid.
Maybe I’m too cautious… But better safe than sorry ;-)