I think if we're going to give shrews blankets, we should just give the shrews blankets, not conditioned on volunteering for this potentially fatal research project.

Nah, do the same thing they did for one of the Venus landers--fasten them with something that melts. Once they have reached an altitude where the temperature won't kill them the tether melts and the blanket falls away. (On Venus they did it to cut a chute because it would have made the lander fall so slowly it would have died before landing.)

However, shrew blankets change the whole scenario to irrelevance--a shrew in a blanket sufficient to keep it safe has very different aerodynamic behavior than a naked shrew.