If you're after a daytime wheelchair then I doubt the home or the LA will pay for it, generally anything the LA/NHS will supply for home use will be available in a care home but the big, personally made chairs aren't (as far as I know) supplied by the home, NHS or LA.
As you say "she needs her own to be measured up for etc" then there maybe a case for saying this is a medical need, but when you say "We would love to be able to move her into the garden on sunny warm days for an hour or so when we visit." are the NHS/LA supposed to pay out £2.5k for a personalised chair just for these occasions?
I don't mean that in a bad way but if the home has said she needs a specialist chair (and here I would ask why is she using "other people's wheelchairs") surely the home has their own chairs suitable for the transport of all the residents they cater for.
I asked today how you get these chairs but the staff just told me the home manager sorts it, no one knew who paid, I'll ask the manager tomorrow if I see her.
If I want a wheelchair in the home (sometime my wife needs one) I just go to the equipment room and get one, some have higher armrests and combined with a thinner cushion that's more suitable for someone who leans a bit, standard cushions and a normal chair are suitable for most people.
I understand your mum is in a care home so I'd expect them to tell you what the alternatives are, and (if like my wife) she's with the home's GP then I'd expect them to do the referral to the appropriate possible funding source, should their be one.
K