Repeated use of the thing helps. I can say from experience that nobody ever expected me to memorize the weights, and only once was I asked to recite the order from 1-103 in front of a group of elementary school kids. The ACS used to hand out little wallet-sized periodic tables as a promo; I think I still have a stack of them somewhere...
Anyway, I've been through every kind of chemistry class offered (being a former Chem Ph.D. student) and I can tell you that nobody beyond first-semester Gen. Chem. feels the need to hide the Periodic Table from you during exams.
If it helps, I'm betting that the PCAT will only deal with Hydrogen through Calcium, and maybe Iron, Bromine, and Iodine. That's 3 less elements to memorize than there are letters in the English alphabet. The general rule is number x2 equals weight, except that Hydrogen is 1. [That's 1.007947 g/mol for fellow chemistry freaks].