Snake substrate is typically okay for them to ingest a bit if it sticks to the mouse. Look at wild snakes, they no doubt ingest all kinds of things while they eat. Dry the mouse off the best you can and take out the water to avoid re-wetting and sticking shouldn't be a problem. If you're especially worried lots of people just throw a paper towel or newspaper square down before feeding and take it out later.
It will not invoke aggression, especially if you feed with a tongs. They will have no reason to associate your hands with food. Just wash your hands before and after feeding so you don't smell like mouse! It's a common myth that tank feeding causes cage aggression, especially if you are feeding correctly (i.e. don't offer food with your fingers).
Feeding out of the enclosure can be stressful. New environment, it's generally not warm, they don't like to be handled after eating. It can definitely lead to food refusal. I don't see a reason to move them, the snake is more likely to eat where they're comfortable.
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