I did setup a test site at http://www.susala.ro/gp-test, with your above suggestions, and seems to mimic OP’s website behaviour. I did tweaked colours, loaded a logo in Site Identity, then a logo in Menu Plus > Sticky, and Sticky Menu Effect to None, Menu Logo Position to Sticky Only.
However, do you think it’s possible to achieve a smooth transition of (elements) of the menu, like in http://www.simonatache.ro? Also, possibility to tweak main logo dimension and Header height independently (like in the example).
Thanks in advance.