While Bitwarden’s Company is based in USA Bitwarden is fully Open-Source and you have the Option to Choose from US and EU servers, technically you can even host it on your own. So depending on how strict you are with the no US product policy this could still be an option. (actually Bitwarden is even listed on the european-alternatives website)

Here also the full list of Password Managers of the European-Alternatives website:

Padloc - Germany - Open Source

Proton Pass - Switzerland - Open Source (well you want to switch away from it, but listing it just for the sake to list every password manager of the EU Alternatives website)

Passbolt - Luxembourg - Open Source

pCloud Pass - Switzerland

heylogin - Germany

Hypervault - Belgium

Password Depot - Germany

uniqkey - Denmark (also useless to list it because it is focused for Companys)

Bitwarden - USA (it’s still listed because you can self-host it) - Open Source

Edit; Formating and Typo