http://www.khmcakes.com/100_1557.jpg

I have a possible client that would like a cake similar to the above, but with only one tier.  Any suggestions on how you would go about getting the flamingos to stand up around the cake?  I'm thinking I'll have a rough time trying to shove two sticks, that are already in place diagonally in the flamingos, into the cake base....

I think that you need to introduce the sticks into the cake, and will be better if the sticks are attached to the cake base to. That will give more support.

AYou can try gluing some fondant balls to the cake board and putting the Flamingo legs in those. For added support, make the flamingos a bit shorter and glue them to the side of the cake.

You could do it as an optical illusion. The legs don't actually support the flamingos, the flamingos are attached to the cake or on a pick which is stuck into the top of the cake, yet close to the edge. The legs are simply there for show, not support. They could just lean in place, or be 'glued', to the sides of the cake. 

Hope this helps.