I would not fly onwards after an obvious bird strike.
There could be various damage not visible from the cockpit, and there could also be prop damage although apparently most birds do just pass through the prop in one piece.
The cowling or the heater air intakes could be blocked.
Also do not raise the landing gear because there could be "stuff" on it which is easier to clean off if it was not moved. The gear doors might also not close properly and could get mangled (hydraulics is very powerful).
Finally there is the risk of prop damage in which case if you read the engine mfg's requirements you will see that a shock load inspection is mandatory (if the prop is actually damaged beyond what can be dressed out conventionally with a file, rather than just paint scratched).