Well, 'go dutch' is not quite right for this sentence; it usually applies to people purchasing for themselves, I think. I am not familiar with 'club together'; perhaps it is BrE? A simple 'get together' would be sufficient. We also 'throw our money into a hat' or 'all chip in' to buy a gift.
As I've understood "go dutch" refers to the situation in which some friends go to a store and everyone buys his/her thing and pay it himself/herself, right?