From the point of view of safety sounds like what you are doing is ideal, even if time consuming. In the NHS you may find a solution closer to home, you may have a look at your wider estates, facilities etc and see what they are doing with theirs, what processes they have and whether they have iso accreditation etc for waste and environment. You may even be surprised that you may find an unrelated department are funded or may help (hopefully)to dispose of these.
when doing a replacement of batteries for an infusion pump batch, 33 which included battery change I found that a local sterile services department had changed their policies to dispose of single use batteries in drills so we combined our efforts here. I can imagine hospital wide, there will be lots of Li-ion batteries going in years to come. Laptops, monitors, phones as well as medical devices so can see it end up being lumped generically into hospital waste, wee, clinnical etc.