Use simple logic. First, we start with the execution vs incarceration:
http://en.wikipedia.org/wiki/Capital_punishment_in_the_United_States
So we now see that very few are executed, so the burden of execution vs incarceration is negligable.
The people who perform the executions take time away from their normal jobs to perform the execution...so they are not adding to the number of people on the government dole. Even if they are, for some reason, the 100 or so people spread across the nation pale in comparison to the approx 100,000 employees of the IRS (depending on the year it is higher or lower).
The guards already guard the prison. The cooks already cook at the prison. On and on. The small number of people to be executed do not change in any realistic way.
There is a barely noticable cost increase for execution as well. Nothing worthy of calling "big government".
So. Basic logic says capital punishment and the size of government are not related.
Your turn. Support your stance they are.