Is it permissible to pay a sum of money to a government employee in exchange for a license, knowing that refusing to do so will lead to the closure of the shop, and that official payment will cost a much larger sum? Is this amount considered a bribe, and what is its expiation?

If taxes, licenses, and the like are illegitimate, then there is no blame upon shop owners for evading or reducing them, even by paying a bribe. This is because it falls under the category of removing injustice, repelling harm, and seeking assistance from whoever can remove the injustice from them.