In answer to teh original question, I have a smart timer (Sonoff mini) fitted to my Viridian charger. The charger was custom built for me by a specialist, inside the box of my broken pod point. I'm sire you could get any competent electrician (i.e doesn't need to be a car charger installer) to fit some sort of device on your feed to the wall charger but obviously at a cost.
However as said above, your box will not consume significant power when not charging - maybe just enough to run an LED or two.
In the first place you really need to understand your usage (- maybe you already do, but this is still a worthwhile exercise). If you have your bills for the past year just compare your usage (not the cost) at the same time of year and see if it has gone up by more than you'd expect for the amount you've charge the car, give or take a bit. I know loads of people are getting horrendous surprises on their bills, even those who don't have an EV, purely as a result of not realising just how much their unit cost has gone up.
Also consider getting on one of the smart tariffs (I'm with Octopus on the Go tariff - see referrer link in my signature below - others use OVO and there are more) which gives you overnight cheap rate (often 4 hours) during which you can charge the car (and do other stuff) at a very cheap rate (in my case at the moment mine is less than a quarter of my day rate) - though you will need a mechanism to turn the charger on and off automatically at the beginning and end of that period.
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