When you've got a full inventory, the only way to use an elixir is to sell an item, buy the elixir, consume it, and buy the original item back. Boots are good for this, since they're cheap, but it's still inconvenient.
A more convenient solution would be to add a 0 cost recipe based on elixirs. First you click on the elixir, then select the "Consume Elixir" recipe that it builds, then you buy the recipe and the elixir's buff is immediately applied.
Attached are a couple images demonstrating what I mean.