For a picklist you have to custom code (see below), however custom relationships you create can have a filter so based on some information on the current record you can filter the result returned by the lookup. Not a picklist, but it might provide a workaround for you. You can expermiment with creating formula fields on the current record that reference $User and $Profile to try to then filter by the formula field.

An idea for how a VF page might work where you don't have to re-write the entire edit page: You'll have to create a VF page (or Lightning Component). If you create a VF page you could put the VF page into the standard page layout. This does cause one usability issue - on the detail page view you'll have your editable user lookup/picklist which you'll have to decide if it gets it's own edit/save mechanism (think about how you change the owner of a record, it's not on the edit page). And on the edit page view you will not see this VF page (i.e. field) at all. But on the view page it will look just like a field, only it will be a tiny VF page. Also you'll want the VF page to do a pop-up when you click the "change" or "edit" mechanism because otherwise the picklist will scroll out of view in the iframe which holds the VF page on the detail page - make sense?