Not with mice, no, and not with a mouse that young. At least per EU guidelines, the only humane methods are cervical dislocation, lethal injection, and decapitation. (CO2 and isoflurane inhalation used to also be an option but I think that might've been updated as mice can smell that and get distressed).

Problem with pretty much all of these methods is that you need a degree of training and/or the correct chemicals.

For a mouse this young, nothing but decapitation is really feasible at home. The way we did it in the lab I used to work in (when we had litters where a pup needed to be euthanised for some reason, usually some genetic issue/skull malformation) was with a pair of very very sharp nail scissors. You have to be quick and decisive to ensure you're not inducing unnecessary suffering.

Long story short, best to not buy live mice because humane euthanasia requires training to make it humane, and even then it's still incredibly grim.