Don't blame yourself for what happened to Dani. Your actions were that of a responsible owner: getting the vaccinations and microchip.
Here micro chipping is very common and is something recommended to owners... but I have never heard of someone sedating an animal to do it. It's not likely that the microchip killed Dani. It definitely could have been a reaction to the sedation, or even to the vaccinations themselves. Unfortunately, its impossible to know what happened.
Blood work may have shown if there were abnormalities with certain organ functions, but its not a guarantee against adverse reactions.
Micro chipping is safe to do at the same time as vaccinations, as it is simply a subcutaneously placed chip. Animals can feel some discomfort as the needle gauge is quite large, but again, I've never heard of a vet sedating due to this.
I am terribly sorry for your loss