The switch does add some resistance. The DMM does also add some resistance, even with short and heavy gauge test leads. I.e. if you include both the switch and the DMM in the circuit you will always get to low a reading.
With only the DMM the reading might be either too low or high, depending on DMM+leads and switch.
The current will also depend on the battery, on a freshly charged battery you will get somewhere between 3.4 and 3.8 volt at 5 ampere, depending on manufacturer and protection circuit. Some curves can be see in my overload current trip test.
I did post some measurements of DMM, leads and switches, see post #21 in this thread.