I don't know if any present it as a FAT/USB or even MTP/PTP (as much as they annoy me they are designed to be a middleman for various protocols and file systems).https://www.amazon.com/NDS-Adapter-Plus-Nintendo-Windows-Pc/dp/B00XVO0I36
is the standard I have a PC and want to dump saves (ROMs are a poorly hidden feature) option, though they are getting quite expensive. Most doing such things just use a DS (various wifi and GBA slot methods exist) or nowadays I guess a 3ds with a custom firmware will do it. The DS and DS lite have a DS slot which can eject and insert games whenever you like so if it is a one off thing I might consider buying a broken screen one of those and doing something.
I don't know what read speeds the DS slot has (it is slower than the GBA slot but still not as slow as USB2.0, or at least some games are worried by latency there) but real time might be pushing it unless you use a RAM dumper or have some flash memory as scratch space (commercial DS games went to 4096 megabit/512 megabytes and random reads anywhere in there are possible, hacked games could go far higher and homebrew is another thing entirely, though you might be able to do something with FCSR). Other than flash carts, general and enhanced, I am not aware of any crazy bankswitching or onboard processing of any great merit, though I would look at some of the keyboard teaching games and you do have various games which used the GBA slot for various things and homebrew like http://hackaday.com/2005/07/20/ds-serial-interface/
which did get used by several things.
The save protocol itself is SPI and covered in http://problemkaputt.de/gbatek.htm#dscartridgebackup
(scroll up from there to see some of the encryption info, though the wood dumper is open source so that would probably be a better thing to look at https://code.google.com/archive/p/wooddumper/source/default/source
Main problem ROM will be the pokemon games that came with a pokewalker as they have a selector on the bus to allow the IR communications to work. It is a simple selector but most things were not built to handle it so they have trouble unless people manually bypass the selector on the cart or with some intermediate device.
If you need a donor board to wire into I like the blaze 3 in 1 game selectors, though they too seem to be getting pricey. If you are able to make boards (and do fairly basic surface mount soldering) then replacement DS or DS lite slots are cheap and very easy to come by.
Equally you are far from the first person to consider a device like this (some think it would be cool, some think it would be more legal or some nonsense like that, some want to hack things and some have other reasons), you would be the first one I heard of that completed it should you do such a thing. It is getting easier I guess though as a lot of stuff I would do on a cheap and cheerful pic today would have taken a big boy FPGA not so long ago. Personally I would find a way to dump and present saves as it would be far easier and still allow you to play games on an emulator and then move back to the cart and have it right there and ready to go.