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I might be wrong, but it sounds like you're on Emacs 27. If you can, check out Emacs 28 (or newer). In addition to the existing tab-bar-move-tab (move tab right), it also provides a bindable tab-bar-move-tab-backward command to move a tab left. Emacs 28 also introduces the concept of "tab groups" and a way to keep all tabs but those of the currently active group hidden (see tab-bar-format, and specifically its tab-bar-format-tabs-groups option). It'll still show the names of the other, inactive tab groups, but at least it doesn't show all their contained tabs. You might be able to hide inactive groups altogether using face properties. Regarding the integration with workgroup packages and jumping between groups of tabs: I had a similar issue regarding the integration of the tab bar and Emacs' project.el package (a project could be considered a workgroup I guess), so I wrote the project-tab-groups package to help with that. The elisp is rather simple, you might be able to take inspiration from it and rework it for use with workgroups/perspective/etc. In case you can't upgrade to Emacs 28, you might want to check out the tab-bar-lost-commands package to help with bindable commands for moving tabs forward and backward (and some other goodies).
I might be wrong, but it sounds like you're on Emacs 27. If you can, check out Emacs 28 (or newer). In addition to the existing tab-bar-move-tab (move tab right), it also provides a bindable tab-bar-move-tab-backward command to move a tab left. Emacs 28 also introduces the concept of "tab groups" and a way to keep all tabs but those of the currently active group hidden (see tab-bar-format, and specifically its tab-bar-format-tabs-groups option). It'll still show the names of the other, inactive tab groups, but at least it doesn't show all their contained tabs. You might be able to hide inactive groups altogether using face properties. Regarding the integration with workgroup packages and jumping between groups of tabs: I had a similar issue regarding the integration of the tab bar and Emacs' project.el package (a project could be considered a workgroup I guess), so I wrote the project-tab-groups package to help with that. The elisp is rather simple, you might be able to take inspiration from it and rework it for use with workgroups/perspective/etc. In case you can't upgrade to Emacs 28, you might want to check out the tab-bar-lost-commands package to help with bindable commands for moving tabs forward and backward (and some other goodies).
Finally, I'm not sure I got what you mean by "reorganise tabs by themselves" exactly, but you might want to check out the tab-bar-groups package, which -- among other things -- provides a tab-bar-groups-regroup-tabs command. If you advice other commands that create/move tabs, you could use it to force all tabs of one group to be next to each other at all times.