Not weighing on death penalty but the narrow scope of your statement.
The government still carries out the whole thing. If you claim to want small government but support the death penalty, you're nothing but a liar.
Your statement seems to lead that there is only one govt. That's the fallacy. There are municipalities (cities, townships, maybe even county) govt, then there's state govt., and finally the federal govt.
When people complain about Big Government, they are talking about the federal level. Why would someone want the size of government to be small at the top (federal level)? Because we have the least amount of say at that level, and it has the most reaching effects. The lower you go, the more influence you have. Therefor the thinking is the more power over your lives they should have. Of course conversely, the lower levels of government also effects less and less people.
You might have heard a saying at one point in time, "The states are experiments in democracy." That was supposed to because the people had a fairly good say in what the state did and didn't do (it's a lot easier to petition a government when you need a lot less number of signatures on a petition) and other states can look at it and say "Hey that went pretty well. Maybe I can implement something like that for my people." or "Well that didn't work out very well, guess we won't be going down that road." You have more control (notice I said more, not total control) over what effects you and what works for you (after all the different areas of the country have different needs) and if someone in a different area of the country wants something that ends up being a bad idea, you're not effected.
Death penalty is handled as a state level. The the federal government shouldn't come into play unless they want to take it away from the states to decide for themselves and make it a federal penalty. So yes, you can still believe in a small government at the federal level and still support the death penalty at the state level.