You're getting a bit confused.

The EU has 1 chamber of parliament, it's not bicameral.

It's the UK that has an unelected chamber of parliament, the House of Lords.

In the EU the parliament counts very little. The EU is run by the commission. The EU commission has executive, legislative, and judiciary powers. The Commission is not elected by the people, but it's nominated by the governments of each member state.

So a EU citizen votes for a member of parliament who then chooses a government who then chooses its EU commissioner.

Perfectly democratic, see? (Joking of course)

The EU does not dictate immigration policy at all. The EU counts little, for now. The swamps are at the national levels, and they have the excuse to blame the EU. For example nothing in EU law forced the UK to turn its cities into Pakistan, but they still did it.