How to assign Back Space on Caps Lock key on Ubuntu 26.4 LTS

What to achieve

Assign Back Space on Caps Lock key.
Assign Control on both Windows key and Menu key.

The author uses SKB-KG2WM2 from Sanwa Supply. The foreign reader should be noted that this model is what is called 109-key layout for Japanese language and lacks right Control key.
By these assignment, ring fingers press Alt key(by the default) and little fingers press Control keys naturally. This is according to the concept that modifier keys such as Shift, Control and Alt should be pressed by the opposite hand of the modified character key. For example, when type Control-P, press Control by little finter of left hand and press P by little finter of right hand, when type Control-Q, press Control by little finger of right hand and press Q by little finger of left hand. Though we lost Menu key, it is not a problem for author because it is not used anyway.
Assign Back Space on Caps Lock key is expected to get equivalent feel of Rub Out key on space-cadet flavor keyboards. Speaking strictly, Rub Out key is said that it sends not Back Space(0x08) but Delete(0x7F), so it must be stoick way to assign Delete on Caps Lock key and tweak by stty erase, but go as is.
Turn from Control at the left of A to this assignment makes Emacs and bash(GNU readline exactly, which is operated as frequently as breathing) comfortable extremely, and the author understood that "eh, thay (not sure, academic community in 1970's U.S.?) used this good stuff!"

What applied

According to the result of giving "What to achieve" to Gemini,

gsettings set org.gnome.desktop.input-sources xkb-options "['caps:backspace', 'altwin:meta_win', 'ctrl:swap_lwin_lctl', 'ctrl:menu_rctrl']"

is tried. It results good in terminal and text box on firefox.
Man gsettings(1) only says that

Name
       gsettings - GSettings configuration tool

SYNOPSIS
       gsettings set SCHEMA*[:*PATH] KEY VALUE

This doesn't make sense at all for the author, the name "gsettings" suggests that it affects as long as GNOME applications. Invoke it without prevelege escalation not causes error but works fine, and another user on same host doesn't affected, so it is understood that it doesn't affect system wide but per user. Once invoke, it is kept in next login session, so the setting must be saved on somewhere under home directory.

Turn to speak Emacs in GUI mode. The problems are that

It behaves as 104-key layout, not 109-key layout (location of some of symbol keys are differ).
It obtains M-x by pressing Menu key (though Alt is expected).
It obtains Alt by pressing Alt key. It is quite right but we want Meta.

Feed them Gemini again and got the solution.
Invoke

setxkbmap jp -option "" -option "caps:backspace,altwin:meta_win,ctrl:swap_lwin_lctl,ctrl:menu_rctrl"

in shell and adding

(setq x-alt-keysym 'meta)
(define-key global-map "×" 'execute-extended-command)

on .emacs solves the problem. Of course, the "jp" on argument on setxkbmap should be adjusted up to you.
Unlike gsettings, invoke setxkmap is required on each session, so add it on .bashrc.