I've used permethrin many times but never in this way. Permethrin in its liquid form should never be applied directly to skin or inhaled. If you treat the bed with permethrin, then allow several hours for the solution to evaporate, leaving the pesticide embedded within the fibres of the sheets/mattress, that is probably safe.

However, the best solution is simply don't stay anywhere that has signs of bedbugs! Permethrin is for unavoidable exposure to disease carrying insects (think malaria), there's no need to use it for bedbugs.