btw. with your semi-transparent flacon at the bottom – maybe it is nice to have the burger overlay semi-transparent.

.html_av-overlay-side #top .av-burger-overlay-scroll {
  background-color: rgba(255,255,255,0.7);
  backdrop-filter: blur(10px)
}

#av-burger-menu-ul .only_mobile img {
  filter: drop-shadow(1px 1px 2px #aaa)
}

there is still something wrong:

I’ll try again – please read it carefully
The menu you see above as a text menu and the menu you see in the hamburger menu have the same source but different selectors. So you don’t need to set up a media query to hide the logo in the top text menu. This is particularly advantageous in your case, as given the nature of your text menu, it would certainly be better to display the hamburger menu earlier (1085px) than would be possible at 768px. Your text menu overlaps the main logo long before it displays the hamburger icon.

for having an earlier hamburger icon you only need here a media query setting:

@media only screen and (max-width: 1085px) {
    #top #header .av-main-nav > li.menu-item  {
        display: none !important;
    }
    #top #header .av-burger-menu-main {
        cursor: pointer;
        display: block !important;
    }
}

Remove all media query settings for your only_mobile logo from your Quick CSS, with the exception of the existing global rule.
(This is why it already works for the footer area.)

.only_mobile {
  display: none;
}

you now only have to show it again for the hamburger menue:

#av-burger-menu-ul .only_mobile {
  display: block;
}

thats all.

btw: here are the selectors for your logos inside your menues:
top text menue: #avia-menu .only_mobile
hamburger menue: #av-burger-menu-ul .only_mobile
footer menue: #menu-main-menu .only_mobile

but – you can have both by css– let the animation stay at slide (slide sidewards or slide up/down)
slides comes in by sliding – but buttons fade in: (maybe with delay)

.avia_transform #top .av_fullscreen .avia-slideshow-button,
.avia_transform #top .av_slideshow_full .avia-slideshow-button,
.avia_transform #top .av_fullscreen .avia-slideshow-button-2, 
.avia_transform #top .av_slideshow_full .avia-slideshow-button-2 {
  opacity: 0;
  transform: translate(0,0);
}

.avia_transform #top .av_fullscreen .active-slide .avia-slideshow-button,
.avia_transform #top .av_slideshow_full .active-slide .avia-slideshow-button,
.avia_transform #top .av_fullscreen .active-slide .avia-slideshow-button-2, 
.avia_transform #top .av_slideshow_full .active-slide .avia-slideshow-button-2 {
  visibility: visible;
  animation: caption-top 1s linear 2s 1 forwards;
}

the shortform of animation means:

animation:
	animation-name: caption-top;
	animation-duration: 1s;
	animation-timing-function: linear;
	animation-delay: 2s;
	animation-iteration-count: 1;
	animation-fill-mode: forwards;

if you do not like the delay – just remove or change that value to your delay ( f.e. 1s)

see: https://webers-testseite.de/slider-with-kenburns/
PS: you see – you can even address them differently the first button can fade in – the second one comes sliding in …

yes – the checked status is for having not a fixed header_meta (top bar)

please do clear all cachings

do not rule it by media query but only with the different selectors

you have still this inside your css: – please remove it

@media only screen and (max-width: 767px) {
  .responsive #top .av-main-nav .menu-item {
    display: none;
  }
}

this works great on DEV Tools for your staging page – if you got different selectors or settings for the page itself – then you have to give us the final link

#avia-menu .only_mobile {
  display: none;
}
#footer-page .only_mobile {
  display: none;
} 

Support topic 1:

How do we limit shop filtering to a single subcategory and apply unique filtering to each category or subcategory?

and do not rule it by media query but only with the different selectors:

#avia-menu .only_mobile {
  display: none;
}

and for the footer:

#menu-main-menu .only_mobile {
  display: none;
}
/ *** or ***/
#footer-page .only_mobile {
  display: none;
} 

because the menu on desktop got an ID and the burger menu is inside .av-burger-overlay and got the id: av-burger-menu-ul

by the way: I can understand it in the case of the hamburger menu, because here it is a copy of the main menu.
But you must have created your footer menu as an extra menu. Why did you put the logo there?

The Demo: enfold-business-flat has on all pages a transparent header. !!!
The transparency logo is shown then. So did you have inserted an alternate logo on :

on the rest of your pages you do not see any logo because the default logo for transparency on that logo is white – and all pages of your page have a white color-section on top.