SUMMARY
Currently, as far as I can tell, Yakuake does not have the ability to start a new session (new tab) with a specific profile, unless you were to go into the settings and set that profile as the default.
You can change the profile of a session after the fact, though this does not execute the custom command.
What I'd like to see is an option (perhaps when you right click on the tab bar) to start a new session with a submenu that lets you select the profile for the session (like Konsole has), as well as being able to set keyboard shortcuts for starting a session with a specific profile (which Konsole actually doesn't have either right now).

My use case:
Having an immutable system (Fedora Kinoite) with several containers of other distributions using distrobox or toolbox. I'd have several different Konsole profiles, the default one for the host system and then several profiles that each have "distrobox enter <container>" set as their custom command.
Then, whenever I need a terminal in a specific container I could just quickly open a new Yakuake tab with the corresponding profile.

SOFTWARE/OS VERSIONS
Operating System: Fedora Linux 37
KDE Plasma Version: 5.26.5
KDE Frameworks Version: 5.102.0
Qt Version: 5.15.8
Kernel Version: 6.1.7-200.fc37.x86_64 (64-bit)
Graphics Platform: Wayland