I've been trying to test it myself and it's proven extremely fragile and unlikely to work in practice. Following this up to step 5 should make the data menu work again, but it will vanish all of your save slot data, as well as reset all of your game settings etc.
1. Load the game and then open the developer console.
2. Go to the "Application" tab, and navigate the tree menu to expand "IndexedDB" -> "Trials in Tainted Space" -> "TITS"
3. You should now see a list of entries of various things. Save slots will be named with keys like "TITS_Slot_0" through "TITS_Slot_14"
4. Expand the "value" column wide enough and you will eventually see a copy button. Do this and copy out the whole mess of characters for each slot. Save each one in a separate file.
5. Once you have the save slots copied out, go back to the tree view and select "Trials in Tainted Space", there should be two buttons. Click the delete one.
6. The stuff you've copied out won't work directly - it's compressed and needs to be decompressed. I can't find a decent online LZString decompressor and treat it as UTF-16 to do this, but if you can find one, the result should be the json that is identical to the Save To File savefiles, and should be able to be loaded as one.