The BBC are very protective of their Doctor Who rights and have shut down a number of people doing similar projects. Zazzle is alive to the issue, as about 6 months ago they removed a number of T-shirt patterns from various suppliers after complaints from the BBC. If you want to make money, then this is a business and you should treat it as such. First, approach the BBC or whatever legal entity actually owns the rights to Doctor Who. You will find that in reality parts of Doctor Who are owned by different entities, but the best place to start is the BBC. Eventually, you will very likely need to buy a type of licence which is often locked to a specific geographic location.
Contacting the BBC is the best first step as they will have someone there who is hired to field similar enquiries and will know the licensing issues. The BBC deals with a large number of groups wishing to use Doctor Who property and I would guess they have strict requirements and set licensing agreements. You may need to present a business plan and I doubt the BBC approach will be very flexible, but it is worth a try. Remember, if the BBC must approve of each design, they need to pay someone to do that and so the profits will need to make it worth their time jumping into what will be fiddly work. Unless you can interest the BBC, you are not going anywhere with your idea. I cannot see how the US copy write law would be an impediment if you are properly licensed. Obviously, if you get serious with your idea you will want to speak to a lawyer who knows a thing or two about intellectual property.
Good luck.