menu at the bottom

the normal non mobile Menu will be hard to get. Because the flyout submenus had to slide out from bottom to top. This maybe possible but is a lot of css to change.

the hamburger is up to you – if you go and set it to postition: fixed you can shift it to whereever you like.
f.e.:

.av-burger-menu-main.menu-item-avia-special {
  position: fixed;
  bottom: 0;
  right: 10px;
  padding: 10px;
}

.av-hamburger {
  padding: 15px 8px;
  display: inline;
  background-color: rgba(255,255,255,0.5);
}

if you choose the header style: logo center – navigation below it would be possible.
in this case the navigation will be in an extra container: header_alternate not inside logo container
and set for: Menu Items For Desktop to icon

then try :

.av-burger-menu-main.menu-item-avia-special {
  position: fixed !important;
  bottom: 0;
  left: 10px;
  padding: 10px;
  z-index: 1001 !important;
}

.av-hamburger {
  padding: 15px 8px;
  display: inline;
  background-color: rgba(255,255,255,1);
  box-shadow: 0 0 15px -5px #333
}

#header, #header_main {
  height: 0px !important
}

@media only screen and (max-width:767px) { 
 .responsive #header_main_alternate {
  display: block
 }
}

.container.av-logo-container, 
.header_bg {
  display: none !important;
}

.responsive #main {
  padding-top: 0 !important
}

if you like to have the burger without distance to the bottom set the padding-bottom to zero:

.av-burger-menu-main.menu-item-avia-special {
  position: fixed !important;
  bottom: 0;
  left: 50%;
  -webkit-transform: translateX(-50%);
  transform: translateX(-50%);
  margin-bottom: -3px;
  padding: 10px 10px 0;
  z-index: 1001 !important;
}

do not forget to colorize the active burger:

.av-burger-overlay-active #top .av-hamburger-inner,
.av-burger-overlay-active #top .av-hamburger-inner::before,
.av-burger-overlay-active #top .av-hamburger-inner::after {
 background-color:#a81010;
}