Topic: How can I perform a System Transfer without the touchscreen?

Hey everybody, sorry if I'm posting in the wrong area about this: I'm new so I hope you will understand.

Last week my 3DS was sitting in my back back when my water bottle leaked and got it wet. I put the 3DS in rice and left it there for about a week. Now the only problems it has is the battery light always stays blue, the sound doesn't work, the 3D doesn't work, the circle pad stays south-east, and worst of all: the touchscreen doesn't function. I can see whatever it is showing on the bottom screen, but I'm unable to tap on anything as it simply doesn't work. These means I'm unable to perform a system-transfer to my New 2DS XL as it is required for me to have a working touchscreen to get to the system-transfer area in the settings.

TL;DR
My touchscreen works, but I can't tap on it. I want to perform a system-transfer to my new 2DS but tapping is required to get to the system-transfer area in the settings menu.

So my question is, what do I do now?

I know I can transfer my Nintendo Network ID by calling Nintendo Support, but I don't just want to transfer my eShop titles, I want to transfer saves with them and everything else on my damaged 3DS (besides, I already called them and they told me it would take 2 days, and it's been a week and they still haven't transferred anything). I was going to send in the damaged 3DS for repairs, but it turns out they no longer repair old 3DS consoles.

Anybody have any ideas? Please if you've had a similar issue or know of a fix then share it with me. I have a few Shiny Pokemon on my Virtual Console games and I want them on my new 2DS XL, the only thing holding me back is the touchscreen not functioning.