If you are rightly concerned about chucking expensive absinthe down the drain then consider straining into a shot glass and serve on the side. The roughly...
Just water with absinthe to chill the glass. No water in the shaker - just ice and other ingredients. Ice will melt and dilute other ingredients as part of the throwing process.
Absinthe and water in the glass and then discarded (or poured into shot glass to be enjoyed on the side). All other ingredients in the shaker and then strained into absinthe and water washed glass.