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Maybe get a trail camera? You can set them up so that they take photos/video when there is motion. They therefore don't use much battery. Mine is quite an old model, but lasts weeks on a charge (it has 8 AA batteries for long life). Then again, from 2 metres away, they'd be fairly distant.
How far away is the garden? You might need quite a long lens, which can be quite expensive. most DSLRs have a compatible 70-300mm which are affordable.
When we moved to the Welsh valleys back in the summer, I tried to set up some feeders, and eventually found the right spot that can be seen from the house with binoculars but is sheltered enough to make the songbirds feel safe (we've got resident sparrowhawks, tawny owls, and buzzards here).
It's super busy down there with all sorts: goldfinches, siskins, dunnocks, a nuthatch, long tailed tits, blue tits, so many sparrows, all that. I reckon about 30 birds are relying on the feeders now so it's getting pricey to keep them full!
Anyway my mother-in-law lives in very urban and mostly birdless Boston and really wants to see what's going here, so any recommendations for a fixed stills/video camera setup? My first thought was GoPro but battery life is obviously short. There's no mains power or wifi, and the only fixing point is directly across a 2 metre wide dirt lane from the feeders. Or do I just pick up the cheapest old DSLR and longish lens setup I can find to grab stills from the garden?