Not silly at all.
You definitely need to mix/stir the beer after adding the priming solution, no question. But I think you can gently stir it without mixing up too much sediment. Also, when you stir the beer, this will breakout some of the CO2 and provide some measure of positive pressure coming out of the opening. This will possibly, slightly protect against O2 ingress. Stirring via that small opening in the lid seems better than transferring to a bottling bucket.
Dump in the solution. Put in long stirring spoon (gently) and touch the bottom. Then raise it ~2 inches and gently swirl the beer for maybe 30 seconds. Let it sit for 5-10 minutes, then start bottling. You could probably be a little aggressive about stirring; CO2 breakout is your friend here.
Any yeast stirred up should settle in those 5 minutes. Even if it doesn't, getting excess yeast in the bottle is not a problem. What you definitely don't want is excess priming sugar in a bottle.