In all honesty the buttons need to go for mobile! I don't use them at all on my phone.
The best mobile experience is to not need the buttons and type the codes manually. Yes its cumbersome because BBCODES are not a clean language for lack of a better term. This is not a problem with the editor but a problem with BBCODES.
I have been preaching this for years but we really need to transition to markdown at least for all the basics and reserve the use of [] format codes for customs only! its extremely easy to use a phone/tablet editor with markdown in fact we do it everyday...have a look at github's interface on a phone lately whats missing? everything but the textarea and submit button.
NOTE I said transition...just allow the current default BBCODES to parse into markdown equivalents and then parse it through markdown
If you absolutely need/want quick access to BBCODE icons on mobile the best option is a toggle-able hidden dropdown menu. This adds a second click to the process but its the cleanest way.
P.S. I am working on a new theme and the editor will be its own component and its on my todo list, but I already have put some thought into it