It all depends on what type of centring you mean?
Centred in relation to the viewport irrespective of content on either side of the menu? Or centred in the space between the logo and the end of the line (which would not be central in the viewport).
If you want an item perfectly centred in the viewport (or the middle of your page wrapper) then you cannot have items floated to the left or right of the menu as that would offset the natural centring. The items to the side of the menu would need to be removed from the flow or you would need to allocate an even amount of space to either side of the menu to account for the logo width.
CSS can only centre in the space available.
These days you would usually use flexbox rather than floats but the same conditions mentioned above will apply.
Here’s an old example using flexbox to allocate equal space left and right of a centred menu.
I’m not a fan of duplicate content unless absolutely necessary and it is best to try and utilise what you have rather than creating separate items. Sometimes its not possible if the design is too complicated but a lot of the time you can use the same elements to achieve the smaller screen design especially if you wrap them in the hamburger menu approach.