Yes. This is totally feasible and normal. You don't need a "3-month visa" for the US. As a UK citizen, you just need an approved ESTA to board your flight to the US, (unless you think you'd be ineligible for one, in which case you'll need to get a regular tourist visa and visit a US consulate). It's valid for 2 years, so you don't need a new one on the way back from MBJ.