I saw your question in Automations last night and meant to comment but forgot. You might be able to get something to work but may not work as well as you hope.
Personally, I would not try to make the timer or an automation adjust the slider entity. Instead, I would incorporate the automation to watch for the slider to change and then restart the timer to that value. (It will require a template which I already have written and use something similar myself.) Maybe it could still be set via the automation without updating the timer and restrict the update when changed by a specific user. In the end, counting down the slider may not be optimal.
For the graphical representation of the timer, I recommend the timer-bar-card. It is very customizable and useful. I, personally, would use it along with a browser_mod pop-up that could contain some buttons for options: restart, cancel, add x minutes, etc.