So I've also contacted Microsoft support and my call was directed towards what I take to be a departement dedicated to support on Surface devices. I spoke to someone who first told me to reset the computer entirely... At which point I refused and pointed him to this thread explaining that many of us are in the same situation and cannot understand that the only official responses we are getting seem to assume this is a system or hardware issue when the problem is quite clearly a compatibility issue with the driver.
Three days later this person got back to me and walked me through uninstalling and deleting the driver (from the "device manager" panel) (despite my having explicitely told him I had tried that many times, mind you). This made bluetooth disappear entirely from the list in the device manager. He then had me refresh that page (the logo of that button is a screen with a magnifying glass). Bluetooth reappeared with the Marvell driver again, but upon closer inspection the version is no longer the same. It used to be 15.68.17021.121. It is now 15.68.9112.23 (2016 version).
He also gave me some form of explanation if you want to call it that. It's what most of us have been saying all along: Intel has updated their driver but there can be conflicts with the rest of the system depending on the set up you have. I couldn't get anyting more specific than that out of him. Because it has to do with an Intel component (he mentionned the graphic card in passing...), this issue is at once not present on all Surface devices and not solely resticted to Surface devices. He seemed to say that the people in charge were working on it...
Of course the moment he hung up, bluetooth crashed again with the infamous code 43... He's calling me back on Monday to see if this fix worked. I know what to say at that point.
Meanwhile I'm back to working using the wireless keyboard dongle I purchased... looks like that was a good investment.