Do you always have a shower in the morning? Do you think the BG rise might in fact be due to dawn-effect (with an additional apparent rise in the CGM, potentially for other reasons re calibration, etc.)? I could imagine that getting in the shower first thing might well be the kick that starts the stress hormones going. My dawn effect doesn't register for about 10min after getting up (on the CGM, so it is probably actually happening as soon as I get up.)
I don't have a shower straight after getting up these days - I usual need to persuade/drag my children from their beds, chivvy them into getting ready and then drive them to school, and then am either WFH, so tend to either do some admin so I know what's coming or sit in a 9am meeting if one's scheduled. If I'm going to work I tend to ride my bike so don't bother with a shower before leaving.
As I tend to have a shower later in the day (lunchtime after an exercise loop on the bike if WFH) or at work/in the evening if doing my cycle commute, this means there's no dawn effect in progress so I see that my CGM rise (though I accept we're all different of course!) is transient. I suppose it's possible my actual BG is also rising then falling rapidly back to where it was by the time I'm dressed. If I remember I'll try to stick my hand out of the shower and do a blood test one day to see what it says, though like you the CGM rise is only on the order of a couple of mmol/l so how easily I'll be able to pick that out of the usual noise is hard to tell.