Will the husband be sinful if he submits a forged university degree—which he can obtain from another country for free—in order to acquire a driver's license for his wife, given that the law in the country where he lives requires a university degree for the husband, and he does not possess one?

Obeying the ruler is not obligatory in restricting what is permissible by Sharia, except for a recognized benefit, and no benefit has appeared from requiring a university degree to obtain a driver's license. If there is a need to obtain a license for the wife, and it is not possible except through that from another country, then there is no harm in obtaining it.