Is it permissible to go for 'Umrah if a relative covers the expenses of the hotel, transportation, and entry visa, in exchange for delivering a bag for him (registered on the passport) to evade customs, knowing that he will be the one to bear the customs duties if payment is required?

If what your cousin provides in terms of accommodation, transportation, and visa is a donation in exchange for a donation from you to carry his luggage, or if it is a wage for delivering the luggage to avoid taxes, then both situations are permissible by mutual consent and by defining the services in a way that eliminates ambiguity. There is no blame upon you for carrying the luggage in exchange for him covering these expenses, and if customs duties are imposed and your cousin pays them, the agreement is not forbidden as long as you are not responsible for paying them.