The way I see it, two basic opinions have been expressed here:
a) Since she holds it’s asur, it’ wrong for her to ask you to do it
b) Not everyone has the same rov or the same situation, so while for her it may indeed be asur, for you it’s not, so there’s no reason not to download shiurim for her.
This leaves you with two options:
1) Refuse to do it, on the grounds that she’s a hypocrite, and has no right to ask you to do something which is beneath her to do.
You risk losing or at least damaging a friendship, but who needs a friend who treats you like her Shabbos goy!
2) Happily do her this chessed, either because you accept opinion (b), or, even if you personally agree with opinion (a), out of the understanding that she may look at things from the perspective of opinion (b), and in fact not look down at you, but rather see things in a more open-minded manner than that; she has her p’sak, but you have yours and are doing nothing wrong by using the internet.
You will be able to help her obtain valuable Torah as well as important medical information, and undoubtedly, through chessed, increase your friendship.
The choice is yours.