Welding different alloys of Al together is always a challenge, especially when it is a non-heat treatable, cold worked (3003 H34) to a heat treatable alloy. (319)

It would be difficult to give real advice regarding the process unless I know more details regardingthe application. I will however say the following:

1) GTAW for more control, which you may need when welding a thick walled component to a thin walled component.
2) GMAW for higher production rates, which you may need if you are welding many of the components, especially if you want to automate the process.

Some other issues to considder:

1) The 3003 H34 is a half hard corld worked and stabilised material. When you weld to it, it will reduce the strength in the HAZ. This may be an important consideration when looking at the design of your component. There is no way for you to regain this strength.
2) The 319 is (as far as I could find out) a 5% Si + 3% Cu heat treatable alloy. This component too will weaken in the HAZ, but this may not be important when I considder your component geometry. You can however regain this loss of strength if you re-heat treat it, but then your 3003 material will lose all its strength.
3) Typical filler to weld these together is 4043 or 4047.
4) An autogenous weld would not be advisable, because you will end up with a Si content of around 2.5%, which is a bit close to the highly crack sensitive range of 0.5 - 2% Si.

Hope this helps

Regards
Niekie Jooste